Waring's Problem in Finite Rings

TL;DR

This paper advances understanding of Waring's problem in finite rings by providing sharp results using algebraic and spectral methods, and extends related theorems to finite fields.

Contribution

It introduces new proofs for Waring's problem over finite rings and establishes an analogue of Sárközy's theorem for finite fields.

Findings

Sharp results for Waring's problem in finite rings

New proofs leveraging Artin-Wedderburn and spectral graph theory

An analogue of Sárközy's theorem for finite fields

Abstract

In this paper we obtain sharp results for Waring's problem over general finite rings, by using a combination of Artin-Wedderburn theory and Hensel's lemma and building on new proofs of analogous results over finite fields that are achieved using spectral graph theory. We also prove an analogue of S\'ark\"ozy's theorem for finite fields.