On maximal and minimal hypersurfaces of Fermat type

Abstract

Let ${\mathbb{F}}_q$ be a finite field with $q=p^n$ elements. In this paper, we study the number of ${\mathbb{F}}_q$-rational points on the affine hypersurface $\mathcal{X}$ given by $a_1{x}_1^{d_1}+\cdots +a_s{x}_s^{d_s}=b$, where $b\in {\mathbb{F}}_q^{\ast }$. 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 $\mathcal{X}$ with respect to Weil's bound.

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