Combinatorics of generic 5-degree polynomials

TL;DR

This paper explores the combinatorial structure of generic complex 5-degree polynomials, analyzing the geometric configurations of their critical values and the associated topological cactus structures, revealing a bipartite graph representation.

Contribution

It introduces a novel combinatorial framework connecting polynomial critical values, cactus structures, and bipartite graphs of genus 3, enriching the understanding of polynomial topology.

Findings

Critical values form convex quadrangles or triangles with an interior point.

Inverse images of these critical value configurations are cactus structures with specific oval counts.

Transformations between cactus types lead to a bipartite graph representation of the polynomial space.

Abstract

We consider the space $P$ of generic complex 5-degree polynomials. Critical values of such polynomial, i.e. four points in the complex plane, either are vertices of a convex quadrangle $Q$, or vertices of a triangle $T$ with one point inside $T$. The inverse image of $Q$ is a tree-like connected structure of five ovals (a cactus). The inverse image of $T$ is also a cactus, but of four ovals. Transformations of cacti of the first type into cacti of the second type and vice versa allow one to represent the space $P$ as a ribbon bipartite graph of genus 3.