The Geometry of the Quadratic Formula

The quadratic formula as an isometry between models of the hyperbolic plane.

We study the quadratic formula \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \left(-b\pm\sqrt{b^2-4ac}\right)/2a geometrically, as a map from the space of coefficients to the space of roots for polynomials of degree \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \leq 2. Our main goal is the following theorem: restricted to real polynomials with complex roots, the quadratic formula realizes an isometry from the projective model to the conformal model of the hyperbolic plane.

Geometry of the map Polynomials \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \longrightarrow Roots

Our goal is to understand in detail the geometry of the quadratic formula as a map from coefficients to roots, and from this understanding extract a cool fact: namely that restricted to real coefficients, the relevant geometry includes both a projective and a conformal model of the hyperbolic plane. Here we start the road to viewing the relationship between a polynomial and its roots as a geometric problem by first casting it as a topological one. As our interest lies in the quadratics everything is quite explicit: however just here at the beginning I will talk about general \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n, as it’s my hope that some of these ideas will (with much more work) continue to tell a good story in higher degree.

The fundamental theorem of algebra guarantees every complex nonconstant polynomial has at least one root, or equivalently every degree \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n polynomial has exactly \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n roots (counting multiplicity). One recasting of this as a topological statement about the map \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots:Polynomials\mapsto Their\;Roots, is as follows. Let \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Pol_n\cong\C^\times\times\C^n denote the space of degree \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n polynomials over \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C identified with their coefficients, and for each \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f\in\Pol_n let \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots(f) be its multiset of roots. To describe the space of these multi-sets, its useful recall the notion of symmetric power: the \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n^{th} symmetric power of a space \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} X is the collection of all unordered \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n-tuples, \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^n(X)=X^n/\Sym(n). The map \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon\C^\times\C^n\to\SP^n(\C) is continuous, and factors through projectivization (\newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f and \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} cf have the same roots) to a map sending a monic polynomial to its roots. The result is a continuous bijection; and thus homeomorphism, between the space of polynomials viewed-as-their-coefficients, and the space of polynomials-viewed-as-their-roots.

This familiar setting is not quite where our story will take place however. Instead of projectivizing the space \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C^\times\times\C^n of degree \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n polynomials onto their monic representatives, we will instead include back in the missing polynomials of lower degree, and study the space \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Pol_{\leq n}\cong\C^{n+1}. This at first seems a difficult task, as the roots map no longer lands in a single space but rather in the union of the symmetric powers \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^n(\C)\cup \SP^{n-1}(\C)\cup\cdots\cup \SP^1(\C)\cup\SP^0(\C). However, it is natural to topologize the union of the first \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n symmetric powers of a space \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} X by using the one-point compactification \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} X\cup\{\infty\}: if \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} k<n we represent the unordered \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} k-tuple \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{x_1,\ldots, x_k\} by the unordered \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n-tuple \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{x_1,\ldots, x_k, \infty,\ldots, \infty\}, essentially filling in the correct number of empty slots with the `filler point’ \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \infty.

For polynomials, this amounts to (for example) saying that, viewed in relation to cubic polynomials, a linear polynomial has two roots at infinity. This is actually quite natural, as when we write down families of cubics which in the limit become linear, two of their roots escape all bounded sets in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C. This leads to the following observation: when we topologize the union \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^n(\C)\cup\SP^{n-1}(\C)\cup\cdots via the aforementioned identification with \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^n(\C\cup\{\infty\}), the natural extension of the roots map \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon \Pol_{\leq n}\to\SP^n(\C\cup\{\infty\}) remains continuous. This map again factors through projectivization of the domain to a continuous bijection, and thus homeomorphism from \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^{n} onto \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^n(\CP^1).

As each space of lower degree polynomials is naturally built into the construction, we may take the direct limit of this over inclusions for increasing \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} n to give a third and final topological incarnation of the fundamental theorem:

What does this tell us? Firstly, that there is a uniform, natural way to consider polynomials of lower degree as degenerations of higher degree polynomials, allowing us to union to domains, codomains and root maps for each lower degree into a single morphism \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^n\to\SP^n(\CP^1). Using this, from the viewpoint of coefficients the geometry of polynomials is naturally contained in the geometry of complex projective space, but from the viewpoint of roots the geometry of polynomials is naturally contained in the geometry of symmetric powers of the sphere. The roots map provides a way of passing between these geometries: if \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} X\subset \Pol_{\leq n} is a collection of poylnomials, we may consider both a projective model \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{P}(\mathsf{Coefs}(X)) and a symmetric powers model \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{Roots}(X).

Quadratics

Restricting to (at most) quadratic polynomials, gives the map \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon \CP^2\to\SP^2(\CP^1).
We will write the projectivized coefficients of a polynomial as \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} [a:b:c] and the roots as \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{z,w\}. The discriminant of the polynomial \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f(z)=az^2+bz+c will be denoted \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \delta(f)=b^2-4ac.

In this low degree, an explicit description of the second symmetric power of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1 is useful: \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1)=(\CP^1\times \CP^1)/\Z_2 where \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Z_2 acts by switching the first and second coordinates. We may even draw a useful picture of this, after identifying \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1 with the unit sphere in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \R^3: let \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{pr}\colon\S^2\to I be the map sending a point on the sphere to its height; its \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z coordinate in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} I=[-1,1]. Then the map \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{pr}\times\mathsf{pr}\colon\S^2\times\S^2\to I\times I gives us a kind of degenerate fibration: the preimage of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} (x,y)\in I^2 is generically a torus, consisting of the latitude circle at height \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} x times the latitude at height \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} y.

\newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \S^2\times\S^2 as a degenerate fiber bundle over the square. Generic point preimages are flat tori, the preimage of points along the boundary are circles and corners are points. Together the boundary square has preimage a necklace of 4-spheres onto which \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} T^2\times I^2 is glued.

The diagonal embedding \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta\colon \S^2\hookrightarrow\S^2\times\S^2 is the preimage of the diagonal in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} I\times I, and the \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Z_2 action permuting coordinates acts freely off of here. Along this diagonal the action is trivial, and so geometrically \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\S^2) is a locus cone singularities in the quotient \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\S^2)\cong\CP^2. Subsets of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\S^2) can be profitably understood by `folding’ a corresponding subset of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \S^2\times\S^2 along the diagonal \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \S^2.

Forming the symmetric product \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{SP}^2(\CP^1) as a quotient of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1 by the map exchanging coordinates. (Note that in many of these cartoon drawings it will appear that \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\CP^1) is a boundary of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1). This is of course not true; in reality it is a codimension-2 sphere shaped singular locus.)

It’s instructive to view some important subsets of polynomials in both the roots and coefficient spaces. In the coefficients model, the at-most-linear polynomials show up as the \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1 at infinity with respect to the affine patch centered at \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} [1:0:0]\simeq z^2. In the roots model, this same set identifies with the \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1 worth of points \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{\infty, z\} in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1). This \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1 intersects another important \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1 in the space of quadrics - the set of polynomials with a double root - in a single point (the constant polynomial, which by our convention has a double root at \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \infty). In the roots model, the quadratics with a double root are easy to describe, they are the image of the diagonal embedding \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\CP^1); the singular locus of the orbifold structure on \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1).

The sphere of at most linear polynomials and the sphere of polynomials having a double root meet only in the constant polynomial.

In the coefficient model this set is more difficult to spell out: it is the projective variety cut out by the discriminant \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(\delta)=V(b^2-4ac). To see this is a 2-sphere, we may change variables so that \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} b^2-4ac is diagonal, \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} x^2+y^2+z^2 (all non-degenerate quadratic forms are equivalent over \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C), and note that in the affine patch \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z=-1 we have \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} x^2+y^2=1 in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C^2, whose solution set is a cylinder; together with two points (the north and south poles) at infinity with respect to this patch.

The projective variety \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(b^2-4ac) in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^2 is homeomorphic to a sphere; as can be seen by changing coordinates to \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(x^2+y^2+z^2) and then computing \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(x^2+y^2=1) in an affine patch.

Next we aim to impose some notions of geometry on the space of quadratics. The coefficient model, \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^2, has a natural notion of geometry, with automorphism group \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(3;\C). Note that this is different than what we would get considering only monic degree 2-polynomials (which would be the affine group for \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C^2), and it allows the mixing of linear with quadratic polynomials by sending roots to \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \infty. To understand the geometry of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1), we first look to its orbifold universal cover \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1. This has automorphism group given by \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Aut(\CP^1) on each factor extended by swapping the factors, or \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Z_2\ltimes \PSL(2;\C)\times\PSL(2;\C). The automorphisms of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1) are those elements above which are well defined (and nontrivial) on the quotient: that is, the diagonal elements \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\PSL(2;\C)). This has a nice geometric interpretation: automorphisms of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1) preserve the singular locus which is itself a copy of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1, and any automorphism of this \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1 is admissable. Equivalently, all automorphisms of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1) are induced from automorphisms of the underlying extended complex plane, and then applied to multi-sets of cardinality 2.

Automorphisms of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{SP}^2(\mathbb{CP}^1) must preserve the singular locus, and thus all arise from automorphisms of the underlying space \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1

Finally, it will be useful to return to the coefficients model, and compute the restricted group of symmetries which `play nicely’ with the roots map. As symmetries of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1) must preserve the singular locus, the diagonal \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1 representing double roots, their realization in the coefficients model must preserve the discriminant locus \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(\delta). As \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \delta is a quadratic form on \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C^3, the automorphisms of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^2 preserving \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(\delta) form the complex orthogonal group \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PO(\delta,\C). This is just an incarnation of the irreducible representation \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SL(2,\C)\to\SO(3;\C), taking the \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SL(2,\C) action on the extended complex plane, applying it to unordered 2-tuples, then interpreting those as roots of a polynomial and viewing the action on coefficients. The fact that all orthogonal groups are isomorphic over \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C tells us that \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta,\C) contains subgroups isomorphic to \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(3;\R) and \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(2,1;\R), whose actions we can understand geometrically. With \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta,\C)\cong\SL(2;\C) acting as conformal transformations of the sphere \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(\delta), as expected an \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(3) subgroup acts as rigid rotations of this sphere, and \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(2,1) subgroups act as Möbius transformations preserving some circle on \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} V(\delta). A particular one of these, namely the real points \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta;\R) will be important below.

In root space, the automorphisms of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1) are the diagonal embedding of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2;\mathbb{C}) as we saw above.
In coefficient space, this group appears instead as a complex orthogonal group, \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta,\C)\subset\PSL(3;\C).

It is interesting to note that from this perspective, it is still natural to allow automorphisms mixing quadratic with linear polynomials, but no longer mixing polynomials with distinct roots with those having a double root.

Real Quadratics

If we restrict to polynomials with real coefficients, \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Pol_{\leq 2}(\C) becomes \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Pol_{\leq 2}(\R) and correspondingly the space of projectivized coefficients changes from \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^2 to \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2. It is more work to describe the image under the roots map. As \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{Roots}\colon\CP^2\to\SP^2(\CP^1) is a homeomorphism, that restriction is an embedding of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2 into \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1) is clear.
To understand this embedding, we will realize this \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2 as a certain 2-complex in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1, `folded over’ the diagonal \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\CP^1).

Real quadratics come in two flavors: those with real roots, and those with complex conjugate roots.
In the \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1 cover, polynomials with complex conjugate roots correspond to the collection \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} (z,\overline{z}) as \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z ranges in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C\cup\infty, forming a sphere. This sphere intersects the sphere \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\CP^1) along the great circle \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{(x,x)\mid x\in\R\cup\infty\}, and so in the quotient gets folded along this great circle into a disk. The polynomials with real roots correspond to the collection \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{(x,y)\mid x,y\in\R\cup\infty\}, which forms a torus in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1. This torus also contains the real diagonal \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{(x,x)\mid x\in\R\cup \infty\} as a \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} (1,1) curve and is folded along it onto a Möbius band in the quotient. Thus, in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1 the relevant 2-complex is the union of a sphere and a torus, glued along a circle as in the cartoon below.
In the quotient, the sphere becomes a disk and the torus a Möbius band, glued along their common circle of intersection: this is the \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2 embedded by the roots map.

The preimage of the \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2 of real polynomials in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1), viewed in the orbifold cover \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1 and its quotient, a creased \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2.

Note that this \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2 contains part of the singular locus of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1). It is constructed from \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \S^2\cup_{\S^1}T^2 folded in half along their common \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \S^1, which we saw above to consist of real double roots. To remember this, I will draw a crease along this curve when a picture denotes the image of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2 under the roots map.

The real quadratic polynomials form an \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2 inside of both \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^2\cong \SP^2(\CP^1). In the root-space, this \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2 intersects the singular locus in the circle of polynomials with a double real root.

To calculate the automorphisms of this subset in each the coefficient and roots viewpoints, we find the subgroup which fixes the real points setwise. In the coefficient view this is easy: the smoothly embedded \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2\subset\CP^2 has automorphism group \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(3;\R)<\PSL(3;\C). On the roots side, any automorphism of this creased \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2 must send the crease to itself (to see this, note that all automorphisms of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1) preserve the singular locus, and so the subgroup preserving real polynomials must preserve the intersection of this singular set with the real points, which is precisely the creased circle). This circle represents the real quadratics with a double root \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\R\cup\infty)\subset\Delta(\CP^1) and the automorphisms preserving it setwise are none other than the diagonal embedding of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2;\R). Viewed back on the other side, the automorphisms of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2\subset\CP^2 which preserve the division into polynomials with a double root and those with distinct roots are the intersection of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Aut(\RP^2) with \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Aut(V(\delta)); that is \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta;\R)=\PGL(3;\R)\cap\SO(\delta;\C). Over \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \R, the discriminant \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \delta=b^2-4ac has signature \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} (2,1), so this is a conjugate of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(2,1) in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PGL(3;\R) preserving the divison of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2 into a disk, Möbius band by the discriminant.

The automorphism group of real quadrics in coefficient space is all of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PGL(3;\R). In root space, any automorphism must preserve the singular locus, so the symmetries are reduced to a copy of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2;\R). Back in coefficient space, this subgroup is realized as \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta;\R)<\PSL(3;\R).

This action of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(2,1) is well known - it acts as the isometries of both the Klein (projective) model of the hyperbolic plane when restricted to the disk, and the projective model of de Sitter 1+1 space when restricted to the Möbius band exterior (or anti de Sitter 1+1 space, as the two are isomorphic in this dimension). The \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SL(2;\R) action on the real polynomials in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1) is maybe not as well known, but comes from gluing together two natural actions on the universal cover \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1. Up here, recall that the \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2 of real polynomials is double covered by a more interesting object; the union of a sphere and torus along a circle. The sphere consists of pairs \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} (z,\overline{z}) for \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z\in\CP^1 and the torus of points \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} (x,y) for each coordinate in the extended reals. The subgroup of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Aut(\CP^1\times\CP^1) fixing this sphere a copy of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2,\C) embedded in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Aut(\CP^1\times\CP^1) as \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} A\mapsto (A,\overline{A}) and the subgroup fixing the torus is the real points, \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2;\R)\times\PSL(2;\R).

Automorphisms of the branched cover of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2_\mathsf{Roots} in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1.

Most of the automorphsims of either component individually do not descend to the quotient \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1); to do so an automorphism must preserve the singular locus, which intersects the picture here in the same \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \S^1 common to the sphere and torus. On the sphere, this selects out only conformal automorphisms preserving the equator, which divides \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1 into two disks, each of which when equipped with this \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2;\R) action is a conformal model of the hyperbolic plane (either the Poincare disk or upper half plane, depending if the stereographic projection to \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \C has projection point on, or off the equatorial circle). Thinking of the original \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1 as the boundary of hyperbolic \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} 3-space when equipped with the \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2,\C) action, we can see fixing a circle in the boundary as restricting the possible isometries to those that preserve that hyperbolic plane (which we can view conformally by then projecting onto the upper, lower hemispheres of the ideal boundary if we would like). On the torus, the corresponding picture is similar, but for anti-de Sitter space. Much like the sphere is the ideal boundary of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathbb{H}^3, the torus is the ideal boundary of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{AdS}^3. Fixing certain curves (in our case the (1,1) curve with respect to the decomposition of isometries as \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2;\R)\times\PSL(2;\R) above) correspond to preserving a lower-dimensional anti de Sitter space \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathsf{AdS}^2, which is isomorphic to de Sitter 2-space as a coincidence of low dimensions and has model an open Möbius band. The \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \PSL(2;\R) action on the creased \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2 in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1) is what you get from gluing these two actions together along the common ideal boundary of their spaces, where they agree.

The \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2 of real polynomials is a union of two natural geometries along their ideal boundary.

The Quadratic Formula

We’ve done all the hard work now; having described the domain and codomain of the roots map assigning coefficients in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^2 to their roots in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1) as well as the symmetries relevant to each side.

The complex roots of a real quadratic come in conjguate pairs, and so are represented in root space by the disk \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{z,\overline{z}\} in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SP^2(\CP^1). In coefficient space, the real polynomials with complex roots are the negative cone of the discriminant, represented by the disk \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \P\{\delta<0\} in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2\subset\CP^2. The first of these spaces is naturally a conformal model of the hyperbolic plane (looking in the double cover \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1\times\CP^1, it is a hemisphere of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1 together with the Möbius transformations preserving it), and the second is naturally a projective model (the action of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta,\R)\cong\SO(2,1) is by projective transformations preserving the metric given by the cross ratio). Restricting the root map to this disk gives the map \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon \D_\mathsf{Coefs}^2\to\D_{\mathsf{Roots}}^2, which is equivariant with respect to the action of hyperbolic isometries on each side (that is, if \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} g\in\Isom(\mathbb{H}^2) then \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots(g.f)=g.\Roots(f) where in the first case \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} g acts as an element of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \SO(\delta;\R) and in the second as an element of \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Delta(\PSL(2;\R)).)

The roots map is an intertwiner for the action of isometries on both sides, and is itself an isometry from the coefficients model to the roots model.

It only remains to show that \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots is an isometry. But this is no work at all, thanks to the symmetry we have established along the way. Since \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots is \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Isom(\Hyp^2) equivariant and this action is transitive, \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots is completely determined by its value at any point. Consider \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f=z^2+1 with \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots(f)=\{i,-i\}.
Now let \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Phi be any isometry between these two models of hyperbolic space equivariant with respect to the given group actions. Without loss of generality we may assume that \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Phi(z^2+1)=\{i,-i\} (if not, pre- and post-compose by isometries of the domain / codomain to make it so). But now \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots, \Phi are \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Isom(\Hyp^2)-equivariant maps from \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \D^2_{\mathsf{Coefs}} to \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \D^2_{\mathsf{Roots}} agreeing on the point \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z^2+1, and so they are equal.

To finish it off, we will write this map down in coordinates, as an explicit map from a disk in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \RP^2 to a disk in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \CP^1. This also constitutes a second proof of the statement above, in which you are free to ignore everything up to here in this writeup and just directly compare the formula to the standard conversion from the Klein disk to the Upper Half Plane, for instance as found on Wikiepdia. Starting with a quadratic \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f=az^2+bz+c, we represent it in the space of projectivized coefficients as \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} [a:b:c], and get at its roots via the familiar quadratic formula. \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots:[a:b:c]\mapsto \left\{ \frac{-b\pm\sqrt{b^2-4ac}}{2a}\right\}

To get useful coordinates on the codomain, note that as the roots are always a complex conjugate pair, one must be in the upper half plane and the other in the lower. Thus we may unambigiously select the root in the upper half plane and represent our pair of roots by a single number \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} x+iy for \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} y>0. Noting that \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \delta<0, we may write this as follows, using the convention that \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \sqrt{-} means `the unique positive square root of’ when applied to real numbers. \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon[a:b:c]\mapsto \frac{-b}{2a}+i\frac{\sqrt{4ac-b^2}}{2a} This provides us with useful coordinates on the codomain, and so our next move is to do something similar for the domain. The first obvious choice is the reduction to monic quadratics using the affine patch \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} a=1, which lets us think of the domain as the set of points in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \R^2=\{(b,c)\} with \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} b^2<4c:

\newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon (b,c)\mapsto \frac{-b}{2}+i\frac{\sqrt{4c-b^2}}{2}

The quadratic formula as usually written is a map from a projective model of} \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \mathbb{H}^2 \textbf{as a paraboloid onto the upper half plane.

This is nothing more than the quadratic formula as learned in grade school, but we know from the above that we may interpret this as an isometry between two copies of the hyperbolic plane! The copy in the codomain is familiar; by selecting coordinates given by the root with positive imaginary part we have naturally landed in the upper half plane. But what is the model in the domain? This `paraboloid’ model is in fact the Klein model in disguise - we have just chosen the wrong affine patch; one that runs parallel to the lightcone instead of transverse to it.

projective model of the hyperbolic plane is a paraboloid, hyperboloid disk depending on choice of affine patch.

To reconstruct the more familiar picture, we need to change the affine patch. The following transformation does the job:

\newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \pmat{ a\\b\\c }= \pmat{ \tfrac{w+u}{2}\\v\\\frac{w-u}{2} }

In these coordinates, the discriminant is \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \delta=u^2+v^2-w^2 and the quadratic formula is the map taking the polynomial \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} (w+u)z^2+2vz+(w-u)=0 to its roots:

\newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon [u:v:w]\mapsto\frac{-v}{u+w}+i\frac{\sqrt{w^2-u^2-v^2}}{u+w}

The disk \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \D^2_\mathsf{Coefs} here is fully contained in the affine patch \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} w=1 and so we may take the domain to be the unit disk centered at \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \vec{0} in \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \R^2=\{(u,v)\} giving the expression below, which is the usual transformation from the Klein disk \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{(u,v)\mid u^2+v^2<1\} to the upper half plane \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \{z\mid \mathsf{Im}(z)>0\} up to possibly a reflection of the domain/codomain, depending on your source.

\newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \Roots\colon (u,v)\mapsto \frac{-v}{1+u}+i\frac{\sqrt{1-u^2-v^2}}{1+u}

After a linear change of coordinates, the quadratic formula provides the usual identification of the Klein model with the upper half plane model of hyperbolic space.

Appendix

Here’s the easy lemma that was used in the proof: an equivariant map between two spaces each equipped with a transitive \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} G action is determined by its value at a point.

Denote \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f(x)=\phi(x)=y. Let \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z\in X, and choose \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} g\in G such that \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z=g.x.
Then evaluating \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f, \phi on \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} z gives \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f(z)=f(g.x)=g.f(x)=g.y and similarly \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} \phi(z)=\phi(g.x)=g.\phi(x)=g.y. Thus \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f(z)=\phi(z) so \newcommand{\Roots}{\mathsf{Roots}} \newcommand{\Pol}{\mathsf{Pol}} \newcommand{\SP}{\mathsf{SP}} \newcommand{\Coefs}{\mathsf{Coefs}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\CP}{\mathbb{C}\mathsf{P}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\RP}{\mathbb{R}\mathsf{P}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\PO}{\operatorname{PO}} \newcommand{\D}{\mathbb{D}} \newcommand{\Hyp}{\mathbb{H}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \renewcommand{\S}{\mathbb{S}} f=\phi.