Rational points on complete symmetric hypersurfaces over finite fields

Abstract

For any affine hypersurface defined by a complete symmetric polynomial in $k\ge 3$ 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 $1\le m\le q-3$ and $q$ is odd. A key ingredient in our proof is Segre's classical theorem on ovals in finite projective planes.

15 pages