A Polynomial PDE Criterion for the −n/d Root of the Bernstein-Sato polynomial of Homogeneous Ideals
Abstract
Let I⊆ℂ[x1,…,xn] 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 bI(s). We apply this criterion to the ideal of maximal minors of a generic m× n matrix and obtain the distinguished root −n; combined with local divisibility along determinantal strata, this allows us to obtain the strong monodromy conjecture 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. Keywords. Bernstein-Sato polynomial, monodromy conjecture.
Related
- Zero loci of Bernstein-Sato ideals are not necessarily linear
- On the Multiplicative Thom-Sebastiani Property for Bernstein-Sato Polynomials of Ideals
- A combinatorial criterion for detecting roots of Bernstein-Sato polynomials
- Bernstein--Sato ideals might not be linear
- Bernstein-Sato polynomials of generic Pfaffian ideals
BibTeX
@misc{chen2026polynomial,
title = {{A Polynomial PDE Criterion for the $-n/d$ Root of the Bernstein-Sato polynomial of Homogeneous Ideals}},
author = {Yifan Chen and Huaiqing Zuo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03976},
eprint = {2609.03976},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03976},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Polynomial PDE Criterion for the −n/d Root of the Bernstein-Sato polynomial of Homogeneous Ideals
AU - Yifan Chen
AU - Huaiqing Zuo
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.03976
UR - https://arxiv.org/abs/2609.03976
AB - Let I⊆ℂ[x_1,…,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). We apply this criterion to the ideal of maximal minors of a generic m× n matrix and obtain the distinguished root −n; combined with local divisibility along determinantal strata, this allows us to obtain the strong monodromy conjecture 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. Keywords. Bernstein-Sato polynomial, monodromy conjecture.
ER -