Estimates on the number of [formula omitted]–rational solutions of variants of diagonal equations over finite fields

Abstract

In this paper we study the set of ▪-rational solutions of equations defined by polynomials evaluated in power-sum polynomials with coefficients in ▪. This is done 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 ▪. 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 more general variants of diagonal equations over finite fields.