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

TL;DR

This paper introduces a novel integration concept using Euler characteristic over the projectivization of function germs, linking it to the Alexander polynomial and zeta-function of plane curve singularities.

Contribution

It presents a new approach to express the Alexander polynomial and zeta-function as integrals over the projectivized space of function germs with respect to Euler characteristic.

Findings

Alexander polynomial expressed as an Euler characteristic integral

Zeta-function represented via Euler characteristic integration

New perspective on plane curve singularity invariants

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.