On the number of solutions of two-variable diagonal quartic equations over finite fields
Junyong Zhao, Yang Zhao, Yujun Niu · AIMS Mathematics · 2020
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 $N(x_1^4+x_2^4 = c) = q+O(q^{\frac{1}{2}}).$