Smooth symmetric systems over a finite field and applications

2023/12/15 by Nardo Giménez, Guillermo Matera, Giménez, Nardo +5
Computer Science · Mathematics · #11G25 #14G05 #14G15 #68W40 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2312.09477

openalex publication_date 2023/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the set of common \mathbbF_q-rational solutions of "smooth" systems of multivariate symmetric polynomials with coefficients in a finite field \mathbbF_q. We show that, under certain conditions, the set of common solutions of such polynomial systems over the algebraic closure of \mathbbF_q has a "good" geometric behavior. This allows us to obtain precise estimates on the corresponding number of common \mathbbF_q-rational solutions. In the case of hypersurfaces we are able to improve the results. We illustrate the interest of these estimates through their application to certain classical combinatorial problems over finite fields.

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