An upper bound on the number of F-jumping coefficients of a principal ideal

TL;DR

This paper establishes an upper bound on the number of F-jumping coefficients for principal ideals with isolated singularities in certain rings, linking Jacobian and test ideals in positive characteristic.

Contribution

It introduces a novel upper bound on F-jumping coefficients for principal ideals with isolated singularities, connecting Jacobian and test ideals in characteristic p.

Findings

Upper bound on F-jumping coefficients established

Relation between Jacobian and test ideals proven

Applicable to rings over finite and arbitrary fields of characteristic p

Abstract

We prove a result relating the Jacobian ideal and the generalized test ideal associated to a principal ideal in $R=k[x_1,...,x_n]$ with $[k:k^p]<\infty$ or in $R=k[[x_1,...,x_n]]$ with an arbitrary field $k$ of characteristic $p>0$. As a consequence of this result, we establish an upper bound on the number of $F$-jumping coefficients of a principal ideal with an isolated singularity.