On maximal and minimal hypersurfaces of Fermat type

TL;DR

This paper investigates conditions under which certain Fermat-type hypersurfaces over finite fields attain the maximum or minimum number of rational points allowed by Weil's bound, providing precise criteria for these extremal cases.

Contribution

It establishes necessary and sufficient conditions for Fermat-type hypersurfaces to be maximal or minimal over finite fields, extending classical bounds.

Findings

Derived explicit criteria for maximality of hypersurfaces.

Derived explicit criteria for minimality of hypersurfaces.

Extended Weil's bound to specific Fermat-type hypersurfaces.

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 $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$, where $b\in\mathbb{F}_q^*$. 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 $\mathcal X$ with respect to Weil's bound.