Bernstein-Sato ideals and hyperplane arrangements

TL;DR

This paper explores the relationship between Bernstein-Sato ideals and hyperplane arrangements, establishing criteria for roots of b-functions, and proves conjectures related to hyperplane arrangements and zeta functions.

Contribution

It introduces a criterion linking Bernstein-Sato ideals to roots of b-functions and proves the multivariable n/d conjecture for hyperplane arrangements.

Findings

Roots of b-functions are determined by Bernstein-Sato ideals for certain polynomials.

The set of roots of the b-function of a free hyperplane arrangement is determined by its intersection lattice.

Proves the multivariable n/d conjecture and the strong monodromy conjecture for hyperplane arrangements.

Abstract

We study the relation between zero loci of Bernstein-Sato ideals and roots of b-functions and obtain a criterion to guarantee that roots of b-functions of a reducible polynomial are determined by the zero locus of the associated Bernstein-Sato ideal. Applying the criterion together with a result of Maisonobe we prove that the set of roots of the b-function of a free hyperplane arrangement is determined by its intersection lattice. We also study the zero loci of Bernstein-Sato ideals and the associated relative characteristic cycles for arbitrary central hyperplane arrangements. We prove the multivariable n/d conjecture of Budur for complete factorizations of arbitrary hyperplane arrangements, which in turn proves the strong monodromy conjecture for the associated multivariable topological zeta functions.