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

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 that these ideals are principal, and the zero loci associated to different factorizations are related by a diagonal property. If, additionally, the divisor is a hyperplane arrangement, we obtain nice estimates for the zero locus of its Bernstein–Sato ideal for arbitrary factorizations and show that the Bernstein–Sato ideal attached to a factorization into linear forms is reduced. As an application, we independently verify and improve upon an estimate of Maisonobe 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.