Local m-adic constancy of F-pure thresholds and test ideals

TL;DR

This paper proves that F-pure thresholds and test ideals are locally constant in the m-adic topology for polynomials with isolated singularities, and provides an algorithm to compute F-jumping numbers and test ideals.

Contribution

It establishes local m-adic constancy of F-pure thresholds and test ideals, and introduces a simple algorithm for computing F-jumping numbers and test ideals.

Findings

F-pure thresholds are locally constant in the m-adic topology.

An algorithm for computing F-jumping numbers and test ideals is provided.

Results relate to the ACC conjecture for F-pure thresholds.

Abstract

In this note, we consider a corollary of the ACC conjecture for F-pure thresholds. Specifically, we show that the F-pure threshold (and more generally, the test ideals) associated to a polynomial with an isolated singularity are locally constant in the m-adic topology of the corresponding local ring. As a by-product of our methods, we also describe a simple algorithm for computing all of the F-jumping numbers and test ideals associated to an arbitrary polynomial over an F-finite field.