# A Hilbert-Kunz function with a periodic term that has a given period

## TL;DR

This paper demonstrates that for any positive integer period, there exists a one-dimensional local ring whose Hilbert-Kunz function's periodic term has exactly that period, expanding understanding of periodicity in algebraic functions.

## Contribution

The paper constructs examples of rings with Hilbert-Kunz functions exhibiting any prescribed immediate periodicity, generalizing previous specific cases.

## Key findings

- Existence of rings with Hilbert-Kunz periodic term of any given period
- Explicit construction of rings with specified periodicity
- Extension of known periodicity results to all positive integers

## Abstract

A result of Monsky states that the Hilbert-Kunz function of a one-dimensional local ring of prime characteristic has a term $\phi$ that is eventually periodic. For example, in the case of a power series ring in one variable over a prime-characteristic field, $\phi$ is the zero function and is therefore immediately periodic with period 1. In additional examples produced by Kunz and Monsky, $\phi$ is immediately periodic with period 2. We show that, for every positive integer $\pi$, there exists a ring for which $\phi$ is immediately periodic with period $\pi$.