The Moduli Space of Polynomial Maps of One Complex Variable

Details The Moduli Space of Polynomial Maps of One Complex Variable The Moduli Space of Polynomial Maps of One Complex Variable

2007-08-19

arxiv.org/abs/0708.2512v2

Mathematics

Algebraic Geometry

36pages; Changed terminology a little

Scientific paper

This paper, motivated from complex dynamics and algebraic geometry, studies the moduli space of polynomial maps of $\mathbb{C}$, from the viewpoint of their fixed-point eigenvalues. More concretely, it describes in detail the fiber structure of the map $\Phi_d$ which corresponds each affine conjugacy class of polinomial maps of degree $d$ to the set of its fixed-point eigenvalues. This map $\Phi_d$ is generically finite and has a very beautiful fiber structure. For any set of fixed-point eigenvalues $\lambda$, the cardinality of its fiber $\Phi_d^{-1}(\lambda)$ is exactly computed in finite steps only from the two combinatorial data $I(\lambda)$ and $K(\lambda)$ that are obtained from $\lambda$. The local fiber structure of the map $\Phi_d$ is also completely determined by $I(\lambda)$ and $K(\lambda)$. As a by-product, some problems on combinatorics are provided.

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