On diagonal equations over finite fields

TL;DR

This paper derives explicit formulas for counting solutions of diagonal equations over finite fields, characterizes when solutions are maximal or minimal, and fully describes Fermat type curves in this context.

Contribution

It provides explicit solution counts for diagonal equations with certain restrictions and characterizes extremal cases, advancing understanding of solution distributions over finite fields.

Findings

Explicit formula for solutions of diagonal equations

Necessary and sufficient conditions for maximal/minimal solutions

Complete characterization of Fermat type curves

Abstract

Let $\mathbb{F}_q$ be a finite field with $q=p^t$ elements. In this paper, we study the number of solutions of equations of the form $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$ over $\mathbb{F}_q$. A classic well-konwn result from Weil yields a bound for such number of solutions. In our main result we give an explicit formula for the number of solutions for diagonal equations satisfying certain natural restrictions on the exponents. In the case $d_1=\dots=d_s$, we present necessary and sufficient conditions for the number of solutions of a diagonal equation being maximal and minimal with respect to Weil's bound. In particular, we completely characterize maximal and minimal Fermat type curves.