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 $x_1^3+x_2^3+\cdots +x_n^3+z{x}_{n+1}^3=0$ respectively. Recently, using the generator of ${\mathbb{F}}_q^{\ast }$, 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 $a_1{x}_1^3+a_2{x}_2^3+a_3{x}_3^3=0$ and our formulas are immediately determined by the coefficients $a_1,a_2$ and $a_3$. These extend and improve earlier results.

18 pages