Rational points on complete symmetric hypersurfaces over finite fields

07/22/2020

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by   Jun Zhang, et al.

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For any affine hypersurface defined by a complete symmetric polynomial in k≥ 3 variables of degree m over the finite field 𝔽_q of q elements, a special case of our theorem says that this hypersurface has at least 6q^k-3 rational points over 𝔽_q 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.