Integrals with respect to the Euler characteristic over spaces of functions and the Alexander polynomial

Abstract

We discuss some formulae which express the Alexander polynomial (and thus the zeta-function of the classical monodromy transformation) of a plane curve singularity in terms of the ring of functions on the curve. One of them describes the coefficients of the Alexander polynomial or of zeta-function of the monodromy transformation as Euler characteristics of some explicitly constructed spaces. For the Alexander polynomial these spaces are complements to arrangements of projective hyperplanes in projective spaces. For the zeta-function they are disjoint unions of such spaces. Under the influence of a result by J.Denef and F.Loeser it was understood that this results are connected with the notion of the motivic integration or rather with its version (in some sense a dual one) where the space of arcs is substituted by the space of functions. The aim of this paper is to discuss the notion of the integral with respect to the Euler characteristics (or with respect to the generalized Euler characteristic) over the space of functions (or over its projectivization) and its connection with the formulae for the coefficients of the Alexander polynomial and of the zeta-function of the monodromy transformation as Euler characteristics of some spaces. The paper will be published in Proceedings of the Steklov Mathematical Institute.