A Note on Bernstein-Sato Varieties for Tame Divisors and Arrangements

TL;DR

This paper studies Bernstein-Sato varieties for tame divisors and arrangements, revealing their geometric properties, principal ideal structure in hyperplane arrangements, and improving estimates for generic arrangements.

Contribution

It introduces new results on the structure of Bernstein-Sato ideals for tame divisors, including their codimension, relations between zero loci, and principal ideal property in hyperplane arrangements.

Findings

Zero loci of Bernstein-Sato ideals are purely codimension one.

Zero loci for different factorizations are related by a diagonal property.

Bernstein-Sato ideals for hyperplane arrangements are principal.

Abstract

For strongly Euler-homogeneous, Saito-holonomic, and tame analytic germs we consider general types of multivariate Bernstein-Sato ideals associated to arbitrary factorizations of our germ. We show the zero loci of these ideals are purely codimension one and the zero loci associated to different factorizations are related by a diagonal property. If, additionally, the divisor is a hyperplane arrangement, we show the Bernstein-Sato ideals attached to a factorization into linear forms are principal. As an application, we independently verify and improve an estimate of Maisonobe's regarding standard Bernstein-Sato ideals for reduced, generic arrangements: we compute the Bernstein-Sato ideal for a factorization into linear forms and we compute its zero locus for other factorizations.