Zero loci of Bernstein-Sato ideals

TL;DR

This paper proves a conjecture linking Bernstein-Sato ideals of multivariate polynomials with cohomology support loci, extending classical results connecting b-functions to monodromy eigenvalues in singularity theory.

Contribution

It establishes a new connection between Bernstein-Sato ideals and cohomology support loci, generalizing foundational theorems in singularity and D-module theory.

Findings

Proved the conjecture relating Bernstein-Sato ideals to cohomology support loci.

Extended classical theorems of Malgrange and Kashiwara to multivariate cases.

Established a new framework for understanding polynomial invariants and local system cohomology.

Abstract

We prove a conjecture of the first author relating the Bernstein-Sato ideal of a finite collection of multivariate polynomials with cohomology support loci of rank one complex local systems. This generalizes a classical theorem of Malgrange and Kashiwara relating the b-function of a multivariate polynomial with the monodromy eigenvalues on the Milnor fibers cohomology.