The moduli space of polynomial maps and their fixed-point multipliers

Abstract

We consider the family MPd of affine conjugacy classes of polynomial maps of one complex variable with degree d≥2, and study the map Φd:MPd→Λ˜d⊂Cd/Sd which maps each f∈MPd to the set of fixed-point multipliers of f. We show that the local fiber structure of the map Φd around λ¯∈Λ˜d is completely determined by certain two sets I(λ) and K(λ) which are subsets of the power set of {1,2,…,d}. Moreover for any λ¯∈Λ˜d, we give an algorithm for counting the number of elements of each fiber Φd−1(λ¯) only by using I(λ) and K(λ). It can be carried out in finitely many steps, and often by hand.