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

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 $R_k=\{x^k:x\in {\mathbb{F}}_q^{\ast }\}$. 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.

16 pages. Small additions and typos corrected. We added. at the end, a small subsection comparing our lower bound for Waring numbers with the other 3 lower bounds known