Zida Xiao | Searcharxiv

arXiv2025

The Nakai Conjecture for isolated hypersurface singularities of modality $\le 2$

The well-known Nakai Conjecture concerns a very natural question: For an algebra of finite type over a characteristic zero field, if the ring of its differential operators is generated by the first order derivations, is the algebra regular? And it is natural to extend the Nakai Conjecture to local domains, in this paper, we verify it for isolated hypersurface singularities of modality $\le 2$, this extends the existing works.

math.AG↗

arXiv2024

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↗