A Polynomial PDE Criterion for the $-n/d$ Root of the Bernstein-Sato polynomial of Homogeneous Ideals

Abstract: Let $I\subseteq\mathbb C[x_1,\ldots,x_n]$ be an ideal generated by homogeneous polynomials of a common degree $d$. We give a polynomial partial differential equation criterion guaranteeing that $-n/d$ is a root of the Bernstein--Sato polynomial $b_I(s)$. The criterion is obtained by studying a quotient of the graph $\mathcal D$-module associated with the generators of $I$ and dualizing an auxiliary finite polynomial PDE system by means of the Fischer pairing. We apply the criterion to the ideal of maximal minors of a generic $m\times n$ matrix and obtain the distinguished root $-n$; combined with local divisibility along determinantal strata, this yields the strong monodromy statement in the maximal-minor case. Finally, we prove that the criterion is stable under enlarging the linear span of the generators, adjoining generators in disjoint variables, products satisfying the natural slope condition, and Thom--Sebastiani sums. These stability results provide new classes of homogeneous ideals and polynomials for which the distinguished Bernstein--Sato root can be detected without computing the full Bernstein--Sato polynomial. Keywords. Bernstein-Sato polynomial, monodromy conjecture.