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

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.