From Differential Values to Roots of the Bernstein-Sato Polynomial

TL;DR

This paper explores how the differential values of a cusp singularity determine certain roots of its Bernstein-Sato polynomial, providing detailed results for cusps with multiplicity up to four.

Contribution

It establishes a connection between differential value semimodules and roots of the Bernstein-Sato polynomial, with specific insights for low multiplicity cusps.

Findings

Differential values determine a subset of Bernstein-Sato roots for cusps.

Precise results obtained for cusps with multiplicity n ≤ 4.

Enhanced understanding of the link between singularity invariants and Bernstein-Sato roots.

Abstract

Let $C$ be a cusp in $(\mathbb C^2,\mathbf 0)$ with Puiseux pair $(n,m)$. This paper is devoted to show how the semimodule of differential values of $C$ determines a subset of the roots of the Bernstein-Sato polynomial of $C$. We add more precise results when the multiplicity of the cusp is $n\leq 4$.