The Moduli Space of Rational Maps and Surjectivity of Multiplier Representation

Abstract

In this paper, we first show that the map ΨRatn of the moduli space of rational maps of degree n to ℂn obtained from multipliers at fixed points is always surjective, while the map ΨPolyn of the moduli space of polynomials of degree n to ℂnt 1 defined similarly is never so if n ≥ 4. Next, in the latter case, we give a sufficient condition and a necessary one for points not in the image of ΨPolyn, and give an explicit parametrization for all such points if n = 4 or 5. Also, we show that the preimage of a generic point by ΨPolyn consists of (n − 2)! points.