Density function for the second coefficient of the Hilbert-Kunz function

TL;DR

This paper introduces a density function for the second coefficient of the Hilbert-Kunz function in the context of toric rings, establishing its properties and connections to Ehrhart quasi-polynomials.

Contribution

It constructs a new density function for the second Hilbert-Kunz coefficient and analyzes its properties, including support, continuity, and multiplicativity, extending prior understanding of Hilbert-Kunz invariants.

Findings

Existence of a compactly supported, continuous except at finitely many points density function g_{R, m}

g_{R, m} is multiplicative under Segre products with explicit formulae

Boundedness results for coefficients of Ehrhart quasi-polynomials of rational polytopes

Abstract

We prove that, analogous to the HK density function, (used for studying the Hilbert-Kunz multiplicity, the leading coefficient of the HK function), there exists a $\beta$-density function $g_{R, {\bf m}}:[0,\infty)\longrightarrow {\mathbb R}$, where $(R, {\bf m})$ is the homogeneous coordinate ring associated to the toric pair $(X, D)$, such that $$\int_0^{\infty}g_{R, {\bf m}}(x)dx = \beta(R, {\bf m}),$$ where $\beta(R, {\bf m})$ is the second coefficient of the Hilbert-Kunz function for $(R, {\bf m})$, as constructed by Huneke-McDermott-Monsky. Moreover we prove, (1) the function $g_{R, {\bf m}}:[0, \infty)\longrightarrow {\mathbb R}$ is compactly supported and is continuous except at finitely many points, (2) the function $g_{R, {\bf m}}$ is multiplicative for the Segre products with the expression involving the first two coefficients of the Hilbert polynomials of the rings involved. Here we also prove and use a result (which is a refined version of a result by Henk-Linke) on the boundedness of the coefficients of rational Ehrhart quasi-polynomials of convex rational polytopes.