The number of solutions of cubic diagonal equations over finite fields

Let $p$ be a prime, $k$ be a positive integer, $q=p^k$, and ${\mathbb{F}}_q$ be the finite field with $q$ elements. Let ${\mathbb{F}}_q^{\ast }$ be the multiplicative group of ${\mathbb{F}}_q$, that is ${\mathbb{F}}_q^{\ast }={\mathbb{F}}_q\setminus \{0\}$. In this paper, explicit formulae for the numbers of solutions of cubic diagonal equations $a_1{x}_1^3+a_2{x}_2^3=c$ and $b_1{x}_1^3+b_2{x}_2^3+b_3{x}_3^3=c$ over ${\mathbb{F}}_q$ are given, with $a_i,b_j\in {\mathbb{F}}_q^{\ast }$ $(1\le i\le 2,1\le j\le 3)$, $c\in {\mathbb{F}}_q$ and $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$. Furthermore, by using the reduction formula for Jacobi sums, the number of solutions of the cubic diagonal equations $a_1{x}_1^3+a_2{x}_2^3+\cdots +a_s{x}_s^3=c$ of $s\ge 4$ variables with $a_i\in {\mathbb{F}}_q^{\ast }$ $(1\le i\le s)$, $c\in {\mathbb{F}}_q$ and $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$, can also be deduced.