Rational points on complete symmetric hypersurfaces over finite fields

Abstract

For any affine hypersurface defined by a complete symmetric polynomial in k≥3 variables of degree m over the finite field Fq of q elements, a special case of our theorem says that this hypersurface has at least 6qk−3 rational points over Fq if 1≤m≤q−3 and q is odd. A key ingredient in our proof is Segre’s classical theorem on ovals in finite projective planes.