Zeros of diagonal equations over finite fields

Abstract:

Let $N$ be the number of solutions $({x_1}, \ldots ,{x_n})$ of the equation (1) \[ (1)\quad {c_1}x_1^{{d_1}} + {c_2}x_2^{{d_2}} + \cdots + {c_n}x_n^{{d_n}} = c\] over the finite field ${F_q}$, where ${d_i}|(q - 1),{c_i} \in F_q^*(i = 1, \ldots ,n)$, and $c \in {F_q}$. If \[ \frac {1} {{{d_1}}} + \frac {1} {{{d_2}}} + \cdots + \frac {1} {{{d_n}}} > b \geq 1\] for some positive integer $b$, we prove that ${q^b}|N$. This result is an improvement of the theorem that $p|N$ obtained by B. Morlaye [7] and also by J. R. Joly [3].