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 $\widehat{\Phi}_d:\mathrm{MC}_d\to \widetilde{\Lambda}_d \subset \mathbb{C}^d / \mathfrak{S}_d$ 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 $\widehat{\Phi}_d^{-1}\left(\bar{\lambda}\right)$ for every $\bar{\lambda} \in \widetilde{\Lambda}_d$ except when the fiber $\widehat{\Phi}_d^{-1}\left(\bar{\lambda}\right)$ 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.