A reduction formula for Waring numbers through generalized Paley graphs

TL;DR

This paper introduces a reduction formula linking Waring numbers over finite fields with properties of generalized Paley graphs, enabling the derivation of many explicit values and infinite families of these numbers.

Contribution

It presents a new reduction formula for Waring numbers using generalized Paley graphs and their Cartesian decomposability, expanding methods to compute explicit values.

Findings

Derived a reduction formula for Waring numbers involving generalized Paley graphs.

Identified arithmetic conditions for applying the reduction formula to generate explicit values.

Established infinite families of explicit even Waring numbers using the reduction formula and code characterization.

Abstract

We give a reduction formula for the Waring number $g(k,q)$ over a finite field $\mathbb{F}_q$. By exploiting the relation between $g(k,q)$ with the diameter of the generalized Paley graph $\Gamma(k,q)$ and by using the characterization due to Pearce and Praeger (2019) of those $\Gamma(k,q)$ which are Cartesian decomposable, we obtain the reduction formula $$g(\tfrac{p^{ab}-1}{bc},p^{ab}) = b g(\tfrac{p^a-1}{c},p^a)$$ for $p$ prime and $a,b,c$ positive integers under certain arithmetic conditions. Then, we find some arithmetic conditions to apply the formula above, which allow us to obtain many infinite families of explicit values of Waring numbers. Finally, we use the reduction formula together with the characterization of $2$-weight irreducible cyclic codes due to Schmidt and White (2002) to find infinite families of explicit even values of $g(k,q)$.