# Bernstein-Sato theory for arbitrary ideals in positive characteristic

## TL;DR

This paper extends Bernstein-Sato theory to arbitrary ideals in positive characteristic, establishing connections between generalized Bernstein-Sato polynomials and $F$-jumping numbers, thus broadening the scope of singularity invariants in algebraic geometry.

## Contribution

It generalizes Bernstein-Sato polynomials from principal to arbitrary ideals in positive characteristic, linking them to $F$-jumping numbers.

## Key findings

- Developed generalized Bernstein-Sato polynomials for arbitrary ideals.
- Proved these polynomials relate to $F$-jumping numbers.
- Extended the theory beyond principal ideals.

## Abstract

Musta\c{t}\u{a} defined Bernstein-Sato polynomials in prime characteristic for principal ideals and proved that the roots of these polynomials are related to the $F$-jumping numbers of the ideal. This approach was later refined by Bitoun. Here we generalize these techniques to develop analogous notions for the case of arbitrary ideals and prove that these have similar connections to $F$-jumping numbers.