Reducing the number of equations defining a subset of the $n$-space over a finite field

Stefan Barańczuk

doi:10.5802/afst.1766

Reducing the number of equations defining a subset of the

n

-space over a finite field

Stefan Barańczuk ¹

¹ Collegium Mathematicum, Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 4, 61-614, Poznań, Poland

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 1, pp. 177-182.

Abstract
Abstract

Let $f_{1}, \dots, f_{k}$ be polynomials defining an algebraic set in affine $n$ -space over a finite field. Suppose $k > n$ . We prove that there exists a system of polynomials $g_{1}, \dots, g_{n}$ , each being a linear combination with scalar coefficients of $f_{1}, \dots, f_{k}$ , defining the same algebraic set. In particular, one reduces the number of equations without increasing the total degree. We also have the corresponding result for systems of homogeneous polynomials defining algebraic sets in projective spaces.

Soient $f_{1}, \dots, f_{k}$ des polynômes définissant un ensemble algébrique dans un $n$ -espace affine sur un corps fini. Supposons $k > n$ . Nous montrons qu’il existe un système de polynômes $g_{1}, \dots, g_{n}$ , chacun étant un combinaison linéaire de $f_{1}, \dots, f_{k}$ , avec des coefficients scalaires, définissant le même ensemble algébrique. En particulier, on réduit le nombre d’équations sans augmenter le degré total. Nous avons aussi le résultat correspondant pour des systèmes de polynômes homogènes définissant des ensembles algébriques dans des espaces projectifs.

Received: 2021-07-27
Accepted: 2022-02-15
Published online: 2024-07-26

DOI: 10.5802/afst.1766

Classification: 11G25, 14A25
Keywords: finite fields, algebraic sets, defining polynomials, reduction

Author's affiliations:

Stefan Barańczuk ¹

¹ Collegium Mathematicum, Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 4, 61-614, Poznań, Poland

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Stefan Barańczuk. Reducing the number of equations defining a subset of the $n$-space over a finite field. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 1, pp. 177-182. doi : 10.5802/afst.1766. https://afst.centre-mersenne.org/articles/10.5802/afst.1766/

References
Cited by

[1] David Eisenbud; E. Graham Evans Every algebraic set in $n$ -space is the intersection of $n$ hypersurfaces, Invent. Math., Volume 19 (1973), pp. 107-112 | DOI | Zbl

[2] Uwe Storch Bemerkung zu einem Satz von M. Kneser, Arch. Math., Volume 23 (1972), pp. 403-404 | DOI | Zbl

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