Rational points on complete symmetric hypersurfaces over finite fields

TL;DR

This paper establishes a lower bound on the number of rational points on complete symmetric hypersurfaces over finite fields, extending understanding of their structure and applying classical geometric theorems.

Contribution

It provides a new lower bound for rational points on symmetric hypersurfaces over finite fields, linking algebraic geometry with classical finite geometry results.

Findings

Hypersurfaces have at least 6q^{k-3} rational points under certain conditions.

The proof utilizes Segre's theorem on ovals in finite projective planes.

Results apply to hypersurfaces defined by complete symmetric polynomials.

Abstract

For any affine hypersurface defined by a complete symmetric polynomial in $k\geq 3$ variables of degree $m$ over the finite field $\mathbb{F}_{q}$ of $q$ elements, a special case of our theorem says that this hypersurface has at least $6q^{k-3}$ rational points over $\mathbb{F}_{q}$ if $1\leq m \leq q-3$ and $q$ is odd. A key ingredient in our proof is Segre's classical theorem on ovals in finite projective planes.