The number of solutions of diagonal cubic equations over finite fields

TL;DR

This paper derives explicit generating functions for counting solutions to diagonal cubic equations over finite fields, providing formulas for specific solution counts that extend previous results.

Contribution

The paper offers new explicit formulas for solution counts and generating functions for diagonal cubic equations over finite fields, improving upon earlier work.

Findings

Derived explicit generating functions for solution counts

Provided formulas for solutions of specific cubic equations

Extended previous results on 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.