An inversion $z\mapsto c/z$ mapping $\widehat C_{m,\alpha}$ into $\widehat C_{m,\beta}$ yields proportional equations

Let $m\ge1$ be an integer, let $\alpha,\beta\in\mathbb C\setminus\mathbb R$ (no condition on their moduli), and let $c\in\mathbb C$ with $c\ne0$. For $\gamma\in\mathbb C$ put $P_\gamma(X,Y)=X^m(\gamma+XY)+Y^m(\bar\gamma+XY)$, as in equation (6), and $C_{m,\gamma}=\{z\in\mathbb C:\operatorname{Re}(z^m(|z|^2+\gamma))=0\}$, so that $P_\gamma(z,\bar z)=2\operatorname{Re}(z^m(|z|^2+\gamma))$. Let $\widehat C_{m,\gamma}$ be the closure of $C_{m,\gamma}$ in the Riemann sphere $\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$, and let $J_c\colon\widehat{\mathbb C}\to\widehat{\mathbb C}$ be the inversion $J_c(z)=c/z$ for $z\in\mathbb C\setminus\{0\}$, $J_c(0)=\infty$, $J_c(\infty)=0$.

Assume that $J_c(\widehat C_{m,\alpha})\subseteq\widehat C_{m,\beta}$. Then there is $k\in\mathbb C$ with $k\ne0$ such that, as polynomials in $\mathbb C[X,Y]$,

The left side equals $X^{m+1}Y^{m+1}P_\beta(c/X,\bar c/Y)$: it is the pullback of $P_\beta$ by $z\mapsto c/z$ (with $X$, $Y$ standing for $z$, $\bar z$), with the denominator cleared.

In Theorem 2 an ambient equivalence between the curves is reduced to the forms $cz$, $c/z$, $c\bar z$, $c/\bar z$. This result treats the inversion case: the cleared pullback of $P_\beta$ is proportional to $P_\alpha$, which yields the ratio in (8) and the condition $c^{2m}=\bar\alpha^{2}$ in (3). The hypothesis is an inclusion on the whole sphere, including the point $0$, which $J_c$ sends to $\infty$. Through Theorem 2 it is also used for Theorem 1 and Remark 5.

Formalization Note: the Riemann sphere is the one-point compactification OnePoint ℂ, $\widehat C_{m,\gamma}$ is the topological closure in it, and $J_c$ is defined pointwise on the sphere with the values above at $0$ and $\infty$.