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

Junyong Zhao; Shaofang Hong; Chaoxi Zhu; Junyong Zhao; Shaofang Hong; Chaoxi Zhu

doi:10.3934/math.2020175

AIMS Mathematics

2020, Volume 5, Issue 3: 2710-2731. doi: 10.3934/math.2020175

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Research article

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

1.
Mathematical College, Sichuan University, Chengdu 610064, P. R. China
2.
School of Mathematics and Statistics, Nanyang Institute of Technology, Nanyang 473004, P. R. China

Received: 28 December 2019 Accepted: 10 March 2020 Published: 17 March 2020
MSC : 11T23, 11T24

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 $c\in\mathbb{F}_q^*$.
- diagonal hypersurface,
- rational point,
- finite field,
- Gauss sum,
- Jacobi sum
Citation: Junyong Zhao, Shaofang Hong, Chaoxi Zhu. The number of rational points of certain quartic diagonal hypersurfaces over finite fields[J]. AIMS Mathematics, 2020, 5(3): 2710-2731. doi: 10.3934/math.2020175

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

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