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

Abstract

We consider the family ${\mathrm{MP}}_d$ of affine conjugacy classes of polynomial maps of one complex variable with degree $d\ge 2$, and study the map ${\Phi }_d:{\mathrm{MP}}_d\to {\tilde{\Lambda }}_d\subset {\mathbb{C}}^d/{\mathfrak{S}}_d$ 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 $\bar{\lambda }\in {\tilde{\Lambda }}_d$ is completely determined by certain two sets $\mathcal{I}(\lambda )$ and $\mathcal{K}(\lambda )$ which are subsets of the power set of $\{1,2,\dots ,d\}$. Moreover for any $\bar{\lambda }\in {\tilde{\Lambda }}_d$, 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.

40pages; Revised expression in Introduction a little, and added proofs for some propositions; results unchanged