Bernstein-Sato polynomials in positive characteristic

TL;DR

This paper extends the concept of Bernstein-Sato polynomials to positive characteristic by defining a sequence of such polynomials using D-modules and divided powers Euler operators, linking them to F-jumping exponents.

Contribution

It introduces a new sequence of Bernstein-Sato polynomials in positive characteristic and establishes their equivalence with F-jumping exponents, bridging a gap from characteristic zero theory.

Findings

Defined Bernstein-Sato polynomials in positive characteristic

Established equivalence with F-jumping exponents

Connected D-module theory with singularity invariants

Abstract

In characteristic zero, the Bernstein-Sato polynomial of a hypersurface can be described as the minimal polynomial of the action of an Euler operator on a suitable D-module. We consider the analogous D-module in positive characteristic, and use it to define a sequence of Bernstein-Sato polynomials (corresponding to the fact that we need to consider also divided powers Euler operators). We show that the information contained in these polynomials is equivalent to that given by the F-jumping exponents of the hypersurface, in the sense of Hara and Yoshida.