Quantifying singularities with differential operators

TL;DR

This paper introduces a new numerical invariant for rings in characteristic zero or p>0 that captures properties like singularity and regularity, extending the concept of the F-signature and providing computational tools and applications.

Contribution

It defines a novel invariant that generalizes the F-signature to characteristic zero and positive characteristic, with computed examples and applications to symbolic powers and Bernstein-Sato polynomials.

Findings

Invariant detects singularity and regularity.

Computed examples where F-signature is unknown.

Results on symbolic powers and Bernstein-Sato polynomials.

Abstract

The $F$-signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties. For example, this invariant detects (non)singularity and strong $F$-regularity. However, it is very difficult to compute. Motivated by different aspects of the $F$-signature, we define a numerical invariant for rings of characteristic zero or $p>0$ that exhibits many of the useful properties of the $F$-signature. We also compute many examples of this invariant, including cases where the $F$-signature is not known. We also obtain a number of results on symbolic powers and Bernstein-Sato polynomials.