The monodromy conjecture for plane meromorphic germs

TL;DR

This paper extends the concept of the monodromy conjecture to plane meromorphic germs by defining a topological zeta function and analyzing its poles, revealing a generalized behavior distinct from the holomorphic case.

Contribution

It introduces a topological zeta function for meromorphic germs and demonstrates that its poles satisfy a generalized monodromy conjecture in the plane case.

Findings

Poles of the topological zeta function do not behave as in the holomorphic case.

The poles satisfy a generalized monodromy conjecture.

The study focuses on the plane case of meromorphic germs.

Abstract

A notion of Milnor fibration for meromorphic functions and the corresponding concepts of monodromy and monodromy zeta function have been introduced in [GZLM1]. In this article we define the topological zeta function for meromorphic germs and we study its poles in the plane case. We show that the poles do not behave as in the holomorphic case but still do satisfy a generalization of the monodromy conjecture.