Siyong Tao | Searcharxiv

arXiv2026

On the relations among algebraic models for the MinRank problem

In this article, we study the relations between algebraic models for the MinRank problem, which is a well-known NP-hard problem. We establish connections between the minors modeling, the Kipnis-Shamir modeling, and the support minors modeling, providing insights into their equivalences and differences. Our results contribute to a deeper understanding of the algebraic structures underlying the MinRank problem and may inform future research on efficient algorithms for solving it.

math.AC↗

arXiv2025

Bernstein-Sato functional equations for ideals in positive characteristic

For an ideal of a regular $\cc$-algebra, its Bernstein-Sato polynomial is the monic polynomial of the lowest degree satisfying an Bernstein-Sato functional equation. We generalize the notion of Bernstein-Sato functional equations to the case of ideals in an $F$-finite ring of positive characteristic $p$, and show the relationship between these equations and Bernstein-Sato roots. By applying this theory, we provide an explicit description of Bernstein-Sato roots of a weighted homogeneous polynomial with an isolated singularity at the origin in characteristic $p$. Moreover, we give multiplicative and additive Thom-Sebastiani properties for the set of Bernstein-Sato roots, which prove the characteristic $p$ analogue of Budur and Popa's question.

math.AC↗