Smooth symmetric systems over a finite field and applications

Autor:	Giménez, Nardo, Matera, Guillermo, Pérez, Mariana, Privitelli, Melina
Rok vydání:	2023
Předmět:	Mathematics - Algebraic Geometry Mathematics - Combinatorics Mathematics - Number Theory 68W40 11G25 14G05 14G15
Druh dokumentu:	Working Paper
Popis:	We study the set of common $\mathbb{F}_q$-rational solutions of "smooth" systems of multivariate symmetric polynomials with coefficients in a finite field $\mathbb{F}_q$. We show that, under certain conditions, the set of common solutions of such polynomial systems over the algebraic closure of $\mathbb{F}_q$ has a "good" geometric behavior. This allows us to obtain precise estimates on the corresponding number of common $\mathbb{F}_q$-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. Comment: 37 pages. arXiv admin note: text overlap with arXiv:1510.03721, arXiv:1807.08052
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2312.09477 Zobrazit plný text záznamu View this record from Arxiv