On Lower Bounds for $s$-multiplicities

TL;DR

This paper investigates lower bounds for a family of multiplicity functions on local rings that interpolate between Hilbert-Samuel and Hilbert-Kunz multiplicities, aiming to understand their properties and potential for deforming classical results.

Contribution

It provides new lower bounds for intermediate multiplicities and explores analogies of minimality conjectures in the context of unmixed singular rings.

Findings

Established lower bounds for $s$-multiplicities.

Presented evidence supporting analogies of Watanabe-Yoshida conjectures.

Analyzed behavior of multiplicities in singular rings.

Abstract

A recent continuous family of multiplicity functions on local rings was introduced by Taylor interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicities. The obvious goal is to use this as a tool for deforming results from one to the other. The values in this family which do not match these classic variants however are not known yet to be well-behaved. This article explores lower bounds for these intermediate multiplicities as well as gives evidence for analogies of the Watanabe-Yoshida minimality conjectures for unmixed singular rings.