Bernstein-Sato functional equations for ideals in positive characteristic

TL;DR

This paper extends Bernstein-Sato functional equations to ideals in positive characteristic rings, relating them to roots and providing explicit descriptions for weighted homogeneous polynomials with isolated singularities.

Contribution

It generalizes Bernstein-Sato equations to positive characteristic, linking them to roots and establishing properties analogous to characteristic zero results.

Findings

Explicit description of Bernstein-Sato roots for weighted homogeneous polynomials

Establishment of Thom-Sebastiani properties in positive characteristic

Relation between functional equations and Bernstein-Sato roots in $F$-finite rings

Abstract

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.