Exact solutions to Waring's problem for finite fields

TL;DR

This paper determines the exact Waring function values for specific large exponents over finite fields, advancing understanding of Waring's problem in algebraic number theory.

Contribution

It introduces a new combinatorial method to compute exact Waring function values for certain exponent-field pairs where the exponent is large.

Findings

Exact values of g(k,q) computed for two infinite families

New combinatorial proof technique developed

Results applicable when k is large relative to q

Abstract

The Waring function $g(k,q)$ measures the difficulty of Waring's problem for $k$th powers in the field of $q$ elements. Its calculation seems to be difficult, and many partial results have been published, notably upper bounds for certain regions of the $k$-$q$-plane. In this paper, we compute the exact value of $g(k,q)$ for two infinite families of exponent-field pairs. In these, $k$ is large compared to $q$. We use a new method of proof that is mainly combinatorial in nature.