AIMS Mathematics, 2020, 5(4): 2979-2991. doi: 10.3934/math.2020192

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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

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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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