  We study the set of common F_q-rational zeros of systems of multivariate
symmetric polynomials with coefficients in a finite field F_q. We establish
certain properties on these polynomials which imply that the corresponding set
of zeros over the algebraic closure of F_q is a complete intersection with
"good" behavior at infinity, whose singular locus has a codimension at least
two or three. These results are used to estimate the number of F_q-rational
points of the corresponding complete intersections. Finally, we illustrate the
interest of these estimates through their application to certain classical
combinatorial problems over finite fields.
