On maximal and minimal hypersurfaces of Fermat type

Let F q \mathbb {F}_q be a finite field with alttext="q equals p Superscript n"> = p n encoding="application/x-tex">q=p^n elements. In this paper, we study the number of -rational points on affine hypersurface alttext="script X"> class="MJX-tex-caligraphic" mathvariant="script">X encoding="application/x-tex">\mathcal X given by alttext="a 1 x d Baseline plus midline-horizontal-ellipsis s Super b"> a 1 x d + ⋯ s b encoding="application/x-tex">a_1 x_1^{d_1}+\dots +a_s x_s^{d_s}=b , where alttext="b element-of double-struck q asterisk"> ∈<!-- ∈ <mml:mo>∗<!-- ∗ encoding="application/x-tex">b\in \mathbb {F}_q^* . A classic well-known result Weil yields bound for such points. This paper presents necessary and sufficient conditions maximality minimality respect to Weil’s bound.