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In this paper we study the set of rational solutions of equations defined bypower sums symmetric polynomials with coefficients in a finite field. We dothis by means of applying a methodology which relies on the study of thegeometry of the set of common zeros of symmetric polynomials over the algebraicclosure of a finite field. We provide improved estimates and existence resultsof rational solutions to the following equations: deformed diagonal equations,generalized Markoff Hurwitz type equations and Carlitz's equations. We extendthese techniques to a more general variants of diagonal equations over finitefields.

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