Poles of the topological zeta function for plane curves and Newton polyhedra

TL;DR

This paper investigates how to identify actual poles of the local topological zeta function for nondegenerate plane curves using Newton polyhedra, refining previous methods based on resolution graphs.

Contribution

It provides a new technique to determine actual poles directly from the Newton polyhedron for nondegenerate plane curves, improving pole detection methods.

Findings

Identifies actual poles from Newton polyhedra for nondegenerate plane curves.

Refines pole detection beyond resolution graph methods.

Enhances understanding of topological zeta functions in singularity theory.

Abstract

The local topological zeta function is a rational function associated to a germ of a complex holomorphic function. This function can be computed from an embedded resolution of singularities of the germ. For nondegenerate functions it is also possible to compute it from the Newton polyhedron. Both ways give rise to a set of candidate poles of the topological zeta function, containing all poles. For plane curves, Veys showed how to filter the actual poles out of the candidate poles induced by the resolution graph. In this note we show how to determine from the Newton polyhedron of a nondegenerate plane curve which candidate poles are actual poles.