A function determined by a hypersurface of positive characteristic

TL;DR

This paper introduces a new function associated with hypersurfaces in positive characteristic, unifying Hilbert-Kunz functions and F-signature as special cases, providing a novel perspective on hypersurface invariants.

Contribution

It defines a new function xi(x) that encapsulates key invariants of hypersurfaces in positive characteristic, linking Hilbert-Kunz and F-signature.

Findings

xi(x) recovers Hilbert-Kunz function at one endpoint

xi(x) recovers F-signature at the other endpoint

Provides a unified framework for hypersurface invariants

Abstract

Let R be a formal power series ring over a perfect field k of prime characteristic p, and let m be the maximal ideal of R. Suppose f is a non-zero element in m. In this paper, we introduce a function xi (x) associated with a hypersurface defined on the closed interval [0, 1] in R. The Hilbert-Kunz function and the F-signature of a hypersurface appear as the values of our function xi (x) on the interval s endpoints.