Bernstein-Sato polynomials of generic Pfaffian ideals

Abstract

We determine all roots and their multiplicities in the Bernstein--Sato polynomial of every generic Pfaffian ideal. In the nonmaximal case, the roots lie in explicit rational windows indexed by the rank strata. Integral roots retain their label multiplicities, whereas every nonintegral root is simple, including values with several labels. The proof combines graph-module relations, skew Capelli operators, central characters, and a weighted Mellin integral. We deduce the strong motivic monodromy conjecture for Pfaffian ideals, with pole orders bounded by Bernstein--Sato root multiplicities. For Pfaffians of size $2r$, the largest possible root multiplicity as the matrix size varies is $\lfloor(\sqrt{8r+1}-1)/2\rfloor$. In the nonmaximal case we also prove that the nonunipotent part of the Verdier specialization has zero nilpotent monodromy. Keywords. Bernstein-Sato polynomial, Pfaffian ideal, root multiplicities, strong monodromy conjecture.