Bernstein-Sato theory for singular rings in positive characteristic

TL;DR

This paper extends Bernstein-Sato theory to singular rings in positive characteristic, establishing finiteness, rationality, and connections to Frobenius-based invariants, advancing understanding of singularity invariants in algebra.

Contribution

It introduces a Bernstein-Sato theory for singular rings in positive characteristic, including finiteness and rationality results, and links to Frobenius invariants.

Findings

Finiteness of Bernstein-Sato roots for large classes of singular rings

Rationality of Bernstein-Sato roots in these rings

Connections between roots and Frobenius-based numerical invariants

Abstract

The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Musta\c{t}\u{a}, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic. In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical invariants defined via the Frobenius map. We also obtain a number of new results and simplified arguments in the regular case.