The Moduli Space of Polynomial Maps and Their Holomorphic Indices: I. Generic Properties in the Case of Having Multiple Fixed Points

TL;DR

This paper extends previous work on classifying complex polynomials by their fixed point indices, providing formulas for the generic case when multiple fixed points are present, thus broadening understanding of polynomial conjugacy classes.

Contribution

It introduces new formulas for counting affine conjugacy classes of polynomials with multiple fixed points based on fixed point indices, filling a gap in the classification theory.

Findings

Derived formulas for generic fixed point index collections

Extended classification to polynomials with multiple fixed points

Provided explicit counts for given degrees and fixed point numbers

Abstract

Following the author's previous works, we continue to consider the problem of counting the number of affine conjugacy classes of polynomials of one complex variable when its unordered collection of holomorphic fixed point indices is given. The problem was already solved completely in the case that the polynomials have no multiple fixed points, in the author's previous papers. In this paper, we consider the case of having multiple fixed points, and obtain the formulae for generic unordered collections of holomorphic fixed point indices, for each given degree and for each given number of fixed points.