The number of solutions of diagonal cubic equations over finite fields

Abstract

Let $\mathbb{F}_q$ be a finite field of $q=p^k$ elements. For any $z\in \mathbb{F}_q$, let $A_n(z)$ and $B_n(z)$ denote the number of solutions of the equations $x_1^3+x_2^3+\cdots+x_n^3=z$ and $x_1^3+x_2^3+\cdots+x_n^3+zx_{n+1}^3=0$ respectively. Recently, using the generator of $\mathbb{F}^{\ast}_q$, Hong and Zhu gave the generating functions $\sum_{n=1}^{\infty}A_n(z)x^n$ and $\sum_{n=1}^{\infty}B_n(z)x^n$. In this paper, we give the generating functions $\sum_{n=1}^{\infty}A_n(z)x^n$ and $\sum_{n=1}^{\infty}B_n(z)x^n$ immediately by the coefficient $z$. Moreover, we gave the formulas of the number of solutions of equation $a_1x_1^3+a_2x_2^3+a_3x_3^3=0$ and our formulas are immediately determined by the coefficients $a_1,a_2$ and $a_3$. These extend and improve earlier results.