Density function for the second coefficient of the Hilbert–Kunz function on projective toric varieties

Abstract

We prove that, analogous to the Hilbert–Kunz density function, (used for studying the Hilbert–Kunz multiplicity, the leading coefficient of the Hibert–Kunz function), there exists a $$\beta $$-density function $$g_{R, \mathbf{m}}:[0,\infty )\longrightarrow {\mathbb {R}}$$, where $$(R, \mathbf{m})$$ is the homogeneous coordinate ring associated with the toric pair (X, D), such that $$\begin{aligned} \int _0^{\infty }g_{R, \mathbf{m}}(x)\mathrm{d}x = \beta (R, \mathbf{m}), \end{aligned}$$where $$\beta (R, \mathbf{m})$$ is the second coefficient of the Hilbert–Kunz function for $$(R, \mathbf{m})$$, as constructed by Huneke–McDermott–Monsky. Moreover, we prove, (1) the function $$g_{R, \mathbf{m}}:[0, \infty )\longrightarrow {\mathbb {R}}$$ is compactly supported and is continuous except at finitely many points, (2) the function $$g_{R, \mathbf{m}}$$ 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.