On maximal and minimal hypersurfaces of Fermat type

Let F q \mathbb {F}_q be a finite field with q = p n q=p^n elements. In this paper, we study the number of F q \mathbb {F}_q -rational points on the affine hypersurface X \mathcal X given by a 1 x 1 d 1 + ⋯ + a s x s d s = b a_1 x_1^{d_1}+\dots +a_s x_s^{d_s}=b , where b ∈ F q ∗ b\in \mathbb {F}_q^* . A classic well-known result of Weil yields a bound for such number of points. This paper presents necessary and sufficient conditions for the maximality and minimality of X \mathcal X with respect to Weil’s bound.