The number of rational points of certain quartic diagonal hypersurfaces over finite fields

Abstract

Let $p$ be an odd prime and let $\mathbb{F}_q$ be a finite field of characteristic $p$ with order $q = p^s$. For $f(x_1, \cdots, x_n)\in\mathbb{F}_q[x_1, ..., x_n]$, we denote by $N(f(x_1, \cdots, x_n) = 0)$ the number of $\mathbb{F}_q$-rational points on the affine hypersurface $f(x_1, \cdots, x_n) = 0$. In 1981, Myerson gave a formula for $N(x_1^4+\cdots+x_n^4 = 0)$. Recently, Zhao and Zhao obtained an explicit formula for $N(x_1^4+x_2^4 = c)$ with $c\in\mathbb{F}_q^*: = \mathbb{F}_q\setminus \{0\}$. In this paper, by using the Gauss sum and Jacobi sum, we arrive at explicit formulas for $N(x_1^4+x_2^4+x_3^4 = c)$ and $N(x_1^4+x_2^4+x_3^4+x_4^4 = c)$ with $c\in\mathbb{F}_q^*$.