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

2015/10/13 by Matera, Guillermo, Perez, Mariana, Privitelli, Melina
#11G25 #11T06 #14G05 #14G15 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1510.03721

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.

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