Bernstein-Sato ideals and 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.