Integration with respect to Euler characteristic over the projectivization of the space of functions and the Alexander polynomial of a plane curve singularity

Abstract

We discuss a notion of integration with respect to the Euler characteristic in the projectivization $\P{\cal O}_{\C^n,0}$ of the ring ${\cal O}_{\C^n,0}$ of germs of functions on $C^n$ and show that the Alexander polynomial and the zeta-function of a plane curve singularity can be expressed as certain integrals over $\P{\cal O}_{\C^2,0}$ with respect to the Euler characteristic.