Topological roots of the Bernstein-Sato polynomial of plane curves

TL;DR

This paper explores the topological roots of the Bernstein-Sato polynomial for plane curve singularities, linking them to divisorial valuations and minimal log resolutions, and proves a key case of the Strong Monodromy Conjecture.

Contribution

It introduces a new characterization of topological roots of the Bernstein-Sato polynomial using divisorial valuations and resolves the multiplicity aspect of the Strong Monodromy Conjecture for plane curves.

Findings

Identifies a set of topological roots containing known roots and poles.

Establishes a link between roots and divisorial valuations.

Proves the multiplicity part of the Strong Monodromy Conjecture for n=2.

Abstract

We study a set of topological roots of the local Bernstein-Sato polynomial of arbitrary plane curve singularities. These roots are characterized in terms of certain divisorial valuations and the numerical data of the minimal log resolution. In particular, this set of roots strictly contains both the opposites of the jumping numbers in $(0, 1)$ and the poles of the motivic zeta function counted with multiplicity. As a consequence, we prove the multiplicity part of the Strong Monodromy Conjecture for $n = 2$.