  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.
