Topological Classification of One-Variable Polynomials.
<P>
(a joint work of D.Zvonkine and S.Lando)
<P>
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.
<P>
The enumeration of topological types also leads
to enumeration of some classes of graphs closely
related to polynomials.