On maximal and minimal hypersurfaces of Fermat type

Abstract

Let Fq be a finite field with q = p n elements. In this paper, we study the number of Fq-rational points on the affine hypersurface X given by a1x d1 1 + · · · + asx ds s = b, where b ∈ F ∗ q . A classic well-konwn result from Weil yields a bound for such number of points. This paper presents necessary and sufficient conditions for the maximality and minimality of X with respect to Weil’s bound.