Counting rational points of quartic diagonal hypersurfaces over finite fields

Let ${\mathbb{F}}_q$ be the finite field of order $q$ where $q=p^k$, $k$ is a positive integer and $p$ is an odd prime. Let ${\mathbb{F}}_q^{\ast }$ represent the nonzero elements of ${\mathbb{F}}_q$. For $f(x_1,\cdots ,x_n)\in {\mathbb{F}}_q[x_1,\cdots ,x_n]$, we used $N(f(x_1,\cdots ,x_n)=0)$ to denote the number of ${\mathbb{F}}_q$-rational points of the affine hypersurface $f(x_1,\cdots ,x_n)=0$. In 2020, Zhao et al. obtained the explicit formulae for $N(x_1^4+x_2^4=c)$, $N(x_1^4+x_2^4+x_3^4=c)$ and $N(x_1^4+x_2^4+x_3^4+x_4^4=c)$ over ${\mathbb{F}}_q$, with $c\in {\mathbb{F}}_q^{\ast }$. In this paper, by using Jacobi sums and an analog of the Hasse-Davenport theorem, we arrived at explicit formulae for $N(a_1{x}_1^4+a_2{x}_2^4=c)$ and $N(b_1{x}_1^4+b_2{x}_2^4+b_3{x}_3^4=c)$ with $a_i,b_j\in {\mathbb{F}}_q^{\ast }(1\le i\le 2,1\le j\le 3)$ and $c\in {\mathbb{F}}_q$. Furthermore, by using the reduction formula for Jacobi sums, the number of rational points of the quartic diagonal hypersurface $a_1{x}_1^4+a_2{x}_2^4+\cdots +a_n{x}_n^4=c$ of $n\ge 4$ variables with $a_i\in {\mathbb{F}}_q^{\ast }$ $(1\le i\le n)$, $c\in {\mathbb{F}}_q$ and $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$, can also be deduced. These extended and improved earlier results.