Bernstein--Sato Ideals and the Logarithmic Data of a Divisor

We study a multivariate version of the Bernstein–Sato polynomial, the so-called Bernstein–Sato ideal, associated to an arbitrary factorization of an analytic germ f - f₁···f_r. We identify a large class of geometrically characterized germs so that the D_X,x[s₁,...,s_r]-annihilator of f^s₁¹···f^s_r^r admits the simplest possible description and, more-over, has a particularly nice associated graded object. As a consequence we are able to verify Budur’s Topological Multivariable Strong Monodromy Conjecture for arbitrary factorizations of tame hyperplane arrangements by showing the zero locus of the associated Bernstein–Sato ideal contains a special hyperplane. By developing ideas of Maisonobe and Narvaez-Macarro, we are able to find many more hyperplanes contained in the zero locus of this Bernstein–Sato ideal. As an example, for reduced, tame hyperplane arrangements we prove the roots of the Bernstein–Sato polynomial contained in [−1,0) are combinatorially determined; for reduced, free hyperplane arrangements we prove the roots of the Bernstein–Sato polynomial are all combinatorially determined. Finally, outside the hyperplane arrangement setting, we prove many results about a certain D_X,x-map ∇_A that is expected to characterize the roots of the Bernstein–Sato ideal.