Estimates on the number of 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 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.