On the number of solutions of two-variable diagonal quartic equations over finite fields
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...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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AIMS Press
2020-04-01
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| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/10.3934/math.2020192/fulltext.html |
| Summary: | 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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| ISSN: | 2473-6988 |