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 $Phi_d:mathrm{MP}_dto widetilde{Lambda}_d subset mathbb{C}^d / mathfrak{S}_d$ 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.