On the number of simultaneous solutions of certain diagonal equations over finite fields

TL;DR

This paper provides explicit estimates and existence results for solutions of diagonal equations over finite fields, using geometric analysis, with applications to Waring's problem and solution distribution modulo primes.

Contribution

It introduces new bounds and existence criteria for solutions of diagonal equations over finite fields based on geometric properties.

Findings

Derived explicit solution estimates for diagonal equations

Established existence results for solutions over finite fields

Applied results to Waring's problem and congruence distributions

Abstract

In this paper we obtain explicit estimates and existence results on the number of $\mathbb{F}_q$-rational solutions of certain systems defined by families of diagonal equations over finite fields. Our approach relies on the study of the geometric properties of the varieties defined by the systems involved. We apply these results to a generalization of Waring's problem and the distribution of solutions of congruences modulo a prime number.