Estimates on the number of rational solutions of variants of diagonal equations over finite fields

TL;DR

This paper investigates the number of rational solutions to various diagonal and related equations over finite fields, using geometric methods to improve estimates and establish existence results.

Contribution

It introduces a geometric approach to analyze solutions of symmetric polynomial equations over finite fields, extending to more general variants of diagonal equations.

Findings

Improved estimates on the number of rational solutions

Existence results for deformed diagonal and Markoff Hurwitz equations

Extension of techniques to broader classes of equations

Abstract

In this paper we study the set of rational solutions of equations defined by power sums symmetric polynomials with coefficients in a finite field. We do this by means of applying a methodology which relies on the study of the geometry of the set of common zeros of symmetric polynomials over the algebraic closure of a finite field. We provide improved estimates and existence results of rational solutions to the following equations: deformed diagonal equations, generalized Markoff Hurwitz type equations and Carlitz's equations. We extend these techniques to a more general variants of diagonal equations over finite fields.