Capelli Operators and the Bernstein-Sato polynomials of Minors

Abstract

For every ideal of minors of a generic complex matrix, we determine all roots and their multiplicities in the Bernstein-Sato polynomial. These roots lie in explicit rational intervals indexed by the rank strata. Every nonintegral root is simple, whereas the multiplicity of each integral root equals the number of labels producing that root. We deduce the strong motivic monodromy conjecture, including the inequality relating pole orders to root multiplicities; for minors of size $r$, the largest possible root multiplicity over all matrix dimensions is $\lfloor\sqrt{4r-1}\rfloor$. Keywords. Bernstein-Sato polynomial, determinantal ideal, root multiplicities, strong monodromy conjecture.