  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.
