The Waring's problem over finite fields through generalized Paley graphs

TL;DR

This paper establishes a connection between Waring's numbers over finite fields and the diameters of generalized Paley graphs, providing new exact values, characterizations, and bounds for these numbers.

Contribution

It introduces a novel link between Waring's problem and graph diameters, finds new exact values, and provides bounds for Waring numbers over finite fields.

Findings

Waring's number equals the diameter of generalized Paley graphs when it exists.

Infinite families of exact Waring numbers are identified from Hamming graph characterizations.

Every positive integer can be realized as a Waring number for some non-prime finite field.

Abstract

We show that the Waring's number over a finite field $\mathbb{F}_q$, denoted $g(k,q)$, when exists, coincides with the diameter of the generalized Paley graph $\Gamma(k,q)=Cay(\mathbb{F}_{q},R_k)$ with $R_k=\{x^k : x\in \mathbb{F}_q^*\}$. We find infinite new families of exact values of $g(k,q)$ from a characterization of graphs $\Gamma(k,q)$ which are also Hamming graphs previously proved by Lim and Praeger in 2009. Then, we show that every positive integer is the Waring number for some pair $(k,q)$ with $q$ not a prime. Finally, we find a lower bound for $g(k,p)$ with $p$ prime by using that $\Gamma(k,p)$ is a circulant graph in this case.