Topological Classification of One-Variable Polynomials.

(a joint work of D.Zvonkine and S.Lando)

A polynomial $P$ of degree $n+1$ is a mapping $P:M\to\CP^1$ such that there is a point in the image (infinity'') with only one preimage. Here $M$ is a compact complex curve. Two polynomials $P_1,P_2$ are topologically equivalent if there is a mapping $f:M\to M$ such that $P_1(f)=P_2$. A topological equivalence preserves the critical values of a polynomial and its degeneracy type (the distribution of its critical points in critical values). The number of topological types of polynomials with fixed critical values is finite. We compute the number of topological types of polynomials on the sphere $M=\CP^1$ for all degeneracy classes. This number is computed by means of the Lyashko--Loojenga mapping associating to each polynomial the (unordered) set of its (finite) critical values.

The enumeration of topological types also leads to enumeration of some classes of graphs closely related to polynomials.