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

Abstract

In this paper we study the set of Fq-rational solutions of equations defined by polynomials evaluated in power-sum polynomials with coefficients in Fq. 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 Fq. We provide improved estimates and existence results of Fq-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.