Number of rational points of symmetric complete intersections over a finite field and applications

Abstract

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.

arXiv admin note: text overlap with arXiv:1408.7014, arXiv:1306.1744

References in corpus (2)

• Prime polynomials in short intervals and in arithmetic progressions
• The distribution of factorization patterns on linear families of polynomials over a finite field