S.Gusein-Zade.
The Alexander polynomial of a plane curve singularity
and the ring of functions on it.

(talk at Arnold's seminar, 20.04.99)

In a previous talk (December 1, 1998) there was described an observation
that the zeta-function of the classical monodromy transformation of an
irreducible plane curve singularity $C$ coincides with the Poincare series
of the ring of (germs of) functions on $C$ (i.e., to
$\sum_{i=0}^\infty \dim(W_i/W_{i+1})t^i$ for the natural filtration $W_i$
of the ring of functions on $C$ defined by the order of a function 
considered as a function on a parameter on the curve $C$) and also
a global analogue of this statement for an algebraic curve with one
place at infinity. There were discussed generalisations of this statement
for reducible (i.e., consisting of several branches) plane curve
singularities. There are two formulae for the Alexander polynomial of
several variables for such a curve in terms of the ring of finctions on
the curve. In one of the formulae the coefficients of the Alexander
polynomial are written explicitly as Euler characteristics of certain
spaces (more precisely, of complements to certain hyperplane
arrangements in complex projective spaces).

The result were obtained with A.Campillo and F.Delgado (University
of Valladolid, Spain.