On the number of solutions of two-variable diagonal quartic equations over finite fields

Junyong Zhao; Yang Zhao; Yujun Niu; Junyong Zhao; Yang Zhao; Yujun Niu

doi:10.3934/math.2020192

AIMS Mathematics

2020, Volume 5, Issue 4: 2979-2991. doi: 10.3934/math.2020192

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Research article

On the number of solutions of two-variable diagonal quartic equations over finite fields

1.
School of Mathematics and Statistics, Nanyang Institute of Technology, Nanyang 473004, P. R. China
2.
Mathematical College, Sichuan University, Chengdu 610064, P. R. China
3.
Nanyang Normal University, Nanyang 473061, P. R. China

Received: 16 November 2019 Accepted: 16 March 2020 Published: 20 March 2020
MSC : 11T23, 11T24

Let $p$ be a odd prime number and let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements. In this paper, by using the Gauss sum and Jacobi sum, we give an explicit formula for the number $N(x_1^4+x_2^4 = c)$ of solutions of the following two-variable diagonal quartic equations over $\mathbb{F}_q$: $x_1^4+x_2^4 = c$ with $c\in\mathbb{F}_q^*$. From this result, one can deduce that $N(x_1^4+x_2^4 = c) = q+O(q^{\frac{1}{2}}).$
- diagonal hypersurface,
- rational point,
- finite field,
- Gauss sum,
- Jacobi sum
Citation: Junyong Zhao, Yang Zhao, Yujun Niu. On the number of solutions of two-variable diagonal quartic equations over finite fields[J]. AIMS Mathematics, 2020, 5(4): 2979-2991. doi: 10.3934/math.2020192

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Abstract

Let $p$ be a odd prime number and let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements. In this paper, by using the Gauss sum and Jacobi sum, we give an explicit formula for the number $N(x_1^4+x_2^4 = c)$ of solutions of the following two-variable diagonal quartic equations over $\mathbb{F}_q$: $x_1^4+x_2^4 = c$ with $c\in\mathbb{F}_q^*$. From this result, one can deduce that $N(x_1^4+x_2^4 = c) = q+O(q^{\frac{1}{2}}).$

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