# Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity

## TL;DR

This paper introduces s-multiplicity, a new function that smoothly interpolates between Hilbert-Samuel and Hilbert-Kunz multiplicities, providing a unified framework with applications to ideal closures and toric rings.

## Contribution

It defines s-multiplicity as a continuous interpolation between two classical multiplicities, extending their properties and applications to ideal closures and toric geometry.

## Key findings

- s-multiplicity is continuous in s and matches Hilbert-Samuel and Hilbert-Kunz multiplicities at extremes.
- Proves an Associativity Formula for s-multiplicity generalizing classical formulas.
- Describes s-multiplicity of monomial ideals in toric rings as a volume in real space.

## Abstract

We define a function, called s-multiplicity, that interpolates between Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to Hilbert-Samuel multiplicity for small values of s and is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it has an Associativity Formula generalizing the Associativity Formulas for Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of closures such that if two ideals have the same s-closure then they have the same s-multiplicity, and the converse holds under mild conditions. We describe the s-multiplicity of monomial ideals in toric rings as a certain volume in real space