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

• Main Authors: Junyong Zhao, Shaofang Hong, Chaoxi Zhu
• Format: Article
• Language: English
• Published: AIMS Press 2020-03-01
• Series: AIMS Mathematics
• Subjects: diagonal hypersurface; rational point; finite field; gauss sum; jacobi sum
• Online Access: https://www.aimspress.com/article/10.3934/math.2020175/fulltext.html

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 $c\in\mathbb{F}_q^*$.