The Moduli Space of Rational Maps and Surjectivity of Multiplier Representation Article

First Online: 26 March 2007 Received: 06 September 2006 Revised: 24 January 2007 DOI :
10.1007/BF03321649

Cite this article as: Fujimura, M. Comput. Methods Funct. Theory (2007) 7: 345. doi:10.1007/BF03321649
Abstract In this paper, we first show that the map Ψ_{Rat} _{n} of the moduli space of rational maps of degree n to ℂ^{n} obtained from multipliers at fixed points is always surjective, while the map Ψ_{Poly} _{n} of the moduli space of polynomials of degree n to ℂ^{nt 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 Ψ_{Poly} _{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 Ψ_{Poly} _{n} consists of (n − 2)! points.

Keywords Rational maps fixed points multiplier moduli space

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Authors and Affiliations 1. Department of Mathematics National Defense Academy Yokosuka Japan