Seminar of Algebra

We consider a multivariate generalization of the Bernstein--Sato polynomial due to Budur as well as the associated $\mathscr{D}{X}[s, \dots, s_{r}]$-module $\mathscr{D}{X}[s, \dots, s_{r}] f_{1}^{s_{1}} \cdots f_{r}^{s_{r}}.$ Generalizing techniques of Walther, we show that for a large class of divisors $f = f_{1} \cdots f_{r}$, the annihilator of $f_{1}^{s_{1}} \cdots f_{r}^{s_{r}}$ is generated by the simplest possible elements. As an application, we verify Budur's Topological Multivariable Strong Monodromy Conjecture for tame hyperplane arrangements. Time permitting, we show that for tame and free arrangements, the generalized Bernstein--Sato variety contains several combinatorially determined hyperplanes.