The limit as p -> infinity of the Hilbert-Kunz multiplicity of sum(x_i^(d_i))

TL;DR

This paper investigates the behavior of the Hilbert-Kunz multiplicity for certain polynomial sums over finite fields as the characteristic prime p approaches infinity, revealing a connection to classical trigonometric series.

Contribution

The authors compute the limit of the Hilbert-Kunz multiplicity as p approaches infinity for sums of monomials, linking it to the power series of sec z + tan z.

Findings

Limit of mu as p -> infinity is 1 + coefficient of z^(s-1) in sec z + tan z

Explicit calculation when each d_i equals 2

Connection to classical trigonometric power series

Abstract

Let p be a prime. The Hilbert-Kunz multiplicity, mu, of the element sum(x_i^(d_i)) of (Z/p)[x_1,..., x_s] depends on p in a complicated way. We calculate the limit of mu as p -> infinity. In particular when each d_i is 2 we show that the limit is 1 + the coefficient of z^(s-1) in the power series expansion of sec z + tan z.