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

Abstract

We show that the Waring number over a finite field Fq, denoted as g(k,q), when exists coincides with the diameter of the generalized Paley graph Γ(k,q)=Cay(Fq,Rk) with Rk={xk:x∈Fq∗}. We find infinite new families of exact values of g(k,q) from a characterization of graphs Γ(k,q) which are also Hamming graphs 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 Γ(k,p) is a circulant graph in this case.