The Hilbert-Kunz function for Binomial Hypersurfaces

TL;DR

This paper provides a general iterative formula for the Hilbert-Kunz function of binomial hypersurfaces over fields of positive characteristic, proving the rationality of the Hilbert-Kunz multiplicity and detailing specific cases.

Contribution

It introduces a new closed-form iterative formula for the Hilbert-Kunz function applicable to all binomial hypersurfaces over any positive characteristic field.

Findings

Hilbert-Kunz multiplicity is rational for binomial hypersurfaces.

Explicit formula for the Hilbert-Kunz function of binomial hypersurfaces.

In 1-dimensional cases, the multiplicity is a positive integer.

Abstract

In this article, I give an iterative closed form formula for the Hilbert-Kunz function for any binomial hypersurface in general, over any feild of arbitrary positive characteristic. I prove that the Hilbert-Kunz multiplicity associated to any Binomial Hypersurface over any field of arbitrary positive characteristic is rational. As an example, I also prove the well known fact that for 1-dimensional Binomial Hypersurfaces, the Hilbert-Kunz multiplicity is a positive integer and give a precise account of the integer.