The Moduli Space of Rational Maps and Surjectivity of Multiplier Representation

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.