On a theory of the $b$-function in positive characteristic

TL;DR

This paper develops a theory of the $b$-function in positive characteristic, defining it as an ideal of functions on $Z_p$, proving finiteness and rationality of roots, and connecting it to $D$-modules and test ideals.

Contribution

It introduces a new framework for the $b$-function in positive characteristic using $D$-modules and Frobenius, establishing finiteness and rationality of roots.

Findings

The $b$-function has finitely many roots.

All roots are negative rational numbers.

The $b$-function relates to test ideals and $D$-modules.

Abstract

We present a theory of the $b$-function (or Bernstein-Sato polynomial) in positive characteristic. Let $f$ be a non-constant polynomial with coefficients in a perfect field $k$ of characteristic $p>0.$ Its $b$-function $b_f$ is defined to be an ideal of the algebra of continuous $k$-valued functions on $\mathbb{Z}_p.$ The zero-locus of the $b$-function is thus naturally interpreted as a subset of $\mathbb{Z}_p,$ which we call the set of roots of $b_f.$ We prove that $b_f$ has finitely many roots and that they are negative rational numbers. Our construction builds on an earlier work of Musta\c{t}\u{a} and is in terms of $D$-modules, where $D$ is the ring of Grothendieck differential operators. We use the Frobenius to obtain finiteness properties of $b_f$ and relate it to the test ideals of $f.$