Smooth symmetric systems over a finite field and applications

Abstract

We study the set of common Fq–rational solutions of “smooth” systems of multivariate symmetric polynomials with coeffi-cients in a finite field Fq. We show that, under certain condi-tions, the set of common solutions of such polynomial systems over the algebraic closure of Fqhas a “good” geometric be-havior. This allows us to obtain precise estimates on the corresponding number of common Fq–rational solutions. In the case of hypersurfaces we are able to strengthen the results. We illustrate the interest of these estimates through their application to certain classical combinatorial problems over finite fields.