The Moduli Space of Polynomial Maps and Their Fixed-Point Multipliers

TL;DR

This paper studies the structure of the space of polynomial maps via fixed-point multipliers, providing a detailed description of fiber structures and an algorithm to count elements in each fiber.

Contribution

It introduces a complete description of the local fiber structure of the multiplier map and presents an algorithm to count fiber elements using combinatorial sets.

Findings

Fiber structure determined by sets rac{}() and rac{}()

Algorithm for counting fiber elements based on these sets

Applicable in finitely many steps, often manually

Abstract

We consider the family $\mathrm{MP}_d$ of affine conjugacy classes of polynomial maps of one complex variable with degree $d \geq 2$, and study the map $\Phi_d:\mathrm{MP}_d\to \widetilde{\Lambda}_d \subset \mathbb{C}^d / \mathfrak{S}_d$ which maps each $f \in \mathrm{MP}_d$ to the set of fixed-point multipliers of $f$. We show that the local fiber structure of the map $\Phi_d$ around $\bar{\lambda} \in \widetilde{\Lambda}_d$ is completely determined by certain two sets $\mathcal{I}(\lambda)$ and $\mathcal{K}(\lambda)$ which are subsets of the power set of $\{1,2,\ldots,d \}$. Moreover for any $\bar{\lambda} \in \widetilde{\Lambda}_d$, we give an algorithm for counting the number of elements of each fiber $\Phi_d^{-1}\left(\bar{\lambda}\right)$ only by using $\mathcal{I}(\lambda)$ and $\mathcal{K}(\lambda)$. It can be carried out in finitely many steps, and often by hand.