On the Number of Points of Algebraic Sets over Finite Fields

TL;DR

This paper establishes upper bounds on the number of rational points of algebraic sets over finite fields, including non-definable sets, with special focus on irreducible and complete intersection cases, and constructs extremal examples.

Contribution

It provides new bounds for rational points on algebraic sets over finite fields, especially for non-absolutely irreducible and complete intersections, and describes families reaching these bounds.

Findings

Derived upper bounds for rational points on algebraic sets over finite fields.

Identified better bounds for irreducible but not absolutely irreducible sets.

Constructed families of algebraic sets with maximum or large numbers of rational points.

Abstract

We determine upper bounds on the number of rational points of an affine or projective algebraic set defined over an extension of a finite field by a system of polynomial equations, including the case where the algebraic set is not defined over the finite field by itself. A special attention is given to irreducible but not absolutely irreducible algebraic sets, which satisfy better bounds. We study the case of complete intersections, for which we give a decomposition, coarser than the decomposition in irreducible components, but more directly related to the polynomials defining the algebraic set. We describe families of algebraic sets having the maximum number of rational points in the affine case, and a large number of points in the projective case. Nous d\'eterminons des majorations du nombre de points d'un ensemble alg\'ebrique affine ou projectif, d\'efini sur une extension d'un corps fini par un syst\`eme d'\'equations polynomiales, y compris dans le cas o\`u l'ensemble alg\'ebrique n'est pas d\'efini sur le corps fini lui-m\^eme. Une attention particuli\`ere est port\'ee aux ensemble alg\'ebriques irr\'eductibles mais non absolument irr\'eductibles, pour lesquels nous obtenons de meilleures bornes. Nous \'etudions le cas des intersections compl\`etes, pour lesquelles nous construisons une d\'ecomposition moins fine que la d\'ecomposition en composantes irr\'eductibles, mais plus directement li\'ee aux polyn\^omes qui d\'efinissent l'ensemble alg\'ebrique. Enfin, nous construisons des familles d'ensembles alg\'ebriques atteignant le nombre maximum de points rationnels dans le cas affine, et comportant de nombreux points dans le cas projectifs.