Abstract
In this paper, we first show that the map ΨRat n of the moduli space of rational maps of degree n to ℂn obtained from multipliers at fixed points is always surjective, while the map ΨPoly n 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 ΨPoly n, 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 ΨPoly n consists of (n − 2)! points.
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Fujimura, M. The Moduli Space of Rational Maps and Surjectivity of Multiplier Representation. Comput. Methods Funct. Theory 7, 345–360 (2007). https://doi.org/10.1007/BF03321649
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DOI: https://doi.org/10.1007/BF03321649