Irrational HK multiplicities are possible for trinomial hyper surfaces

TL;DR

This paper explores the possibility of irrational Hilbert-Kunz multiplicities in trinomial hypersurfaces, particularly over fields of characteristic 2, challenging previous assumptions of rationality.

Contribution

It provides a theoretical argument supporting the existence of irrational Hilbert-Kunz multiplicities in trinomial hypersurfaces over characteristic 2 fields.

Findings

Supports the existence of irrational Hilbert-Kunz multiplicities

Focuses on trinomial hypersurfaces in characteristic 2

Provides theoretical reasoning rather than explicit examples

Abstract

A `trinomial hyper surface' is defined in \S 1 below. In this article, I provide a supportive reasoning towards the fact that there can be examples of trinomial hyper surfaces (at least over fields of characteristic 2) for which the corresponding Hilbert-Kunz multiplicity can become irrational.