arXiv subjects

Toshi Sugiyama

Publications and source records attributed to Toshi Sugiyama.

3 recordsLinked to original sources

arXiv2023

The Moduli Space of Polynomial Maps and Their Fixed-Point Multipliers: II. Improvement to the Algorithm and Monic Centered Polynomials

We consider the family $\mathrm{MC}_d$ of monic centered polynomials of one complex variable with degree $d \geq 2$, and study the map $\widehatΦ_d:\mathrm{MC}_d\to \widetildeΛ_d \subset \mathbb{C}^d / \mathfrak{S}_d$ which maps each $f \in \mathrm{MC}_d$ to its unordered collection of fixed-point multipliers. We give an explicit formula for counting the number of elements of each fiber $\widehatΦ_d^{-1}\left(\barλ\right)$ for every $\barλ \in \widetildeΛ_d$ except when the fiber $\widehatΦ_d^{-1}\left(\barλ\right)$ contains polynomials having multiple fixed points. This formula is not a recursive one, and is a drastic improvement of our previous result [T. Sugiyama, The moduli space of polynomial maps and their fixed-point multipliers. Adv. Math. 322 (2017), 132--185] which gave a rather long algorithm with some induction processes.

math.DS↗

arXiv2020

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

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.

math.DS↗

arXiv2017

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

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 $Φ_d:\mathrm{MP}_d\to \widetildeΛ_d \subset \mathbb{C}^d / \mathfrak{S}_d$ 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 $Φ_d$ around $\barλ \in \widetildeΛ_d$ is completely determined by certain two sets $\mathcal{I}(λ)$ and $\mathcal{K}(λ)$ which are subsets of the power set of $\{1,2,\ldots,d \}$. Moreover for any $\barλ \in \widetildeΛ_d$, we give an algorithm for counting the number of elements of each fiber $Φ_d^{-1}\left(\barλ\right)$ only by using $\mathcal{I}(λ)$ and $\mathcal{K}(λ)$. It can be carried out in finitely many steps, and often by hand.

math.AG↗