In a similar way, we can construct a matching on the set $C_\beta(f)
\cap S_r$, and the existence of $n$ distinct roots follows from the combinatorial topology of these two matchings. In the remainder of this article, we study which (pairs of) such matchings can arise from polynomials. We begin by formalizing our study of the matchings which arise from the curves $C_\theta(f)$.

Suppose that $C_\theta(f)$ is nonsingular. By the maximum principle for harmonic functions, the curve $C_\theta(f)$ cannot have any bounded connected components. It follows that every connected component of $C_\theta(f)$ is diffeomorphic to ${{\mathbb R}}$. By Bézout’s theorem, $C_\theta(f)$ cannot meet the quadratic curve $S_r$ in more than $2n$ points. For $r$ sufficiently large, each connected component of $C_\theta(f)$ must meet $S_r$ in at least two points, so we deduce that $C_\theta(f)$ has at most $n$ connected components.

Recall, however, that the curve $C_\theta(f)$ has $2n$ asymptotes, with slopes at angles $(k\pi+\theta)/n$ for $k=0,\ldots,2n-1$. Each connected component approaches at most two of these asymptotes, and it follows that $C_\theta(f)$ has precisely $n$ connected components, each diffeomorphic to ${{\mathbb R}}$, and each approaching two of these asymptotes. Moreover, if $r$ is sufficiently large then each point of $C_\theta(f) \cap 