On diagonal equations over finite fields

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 $a_1{x}_1^{d_1}+\cdots +a_s{x}_s^{d_s}=b$ 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 $d_1=\cdots =d_s$, 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.

17 pages, comments are welcome