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

TL;DR

This paper investigates the number of rational points on symmetric complete intersections over finite fields, establishing properties that lead to precise point estimates and applications to combinatorial problems.

Contribution

It introduces new properties of symmetric polynomials that ensure well-behaved zero sets, enabling accurate estimates of rational points over finite fields.

Findings

Complete intersections have singular loci of high codimension.

Derived bounds for the number of rational points.

Applications to classical combinatorial problems.

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.