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...
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| Format: | Article |
| Language: | English |
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AIMS Press
2020-04-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/10.3934/math.2020192/fulltext.html |
| _version_ | 1819063234500493312 |
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| author | Junyong Zhao Yang Zhao Yujun Niu |
| author_facet | Junyong Zhao Yang Zhao Yujun Niu |
| author_sort | Junyong Zhao |
| collection | DOAJ |
| description | 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 $N(x_1^4+x_2^4=c)=q+O(q^{\frac{1}{2}}).$ |
| first_indexed | 2024-12-21T15:11:26Z |
| format | Article |
| id | doaj.art-bdac80ab8f06465aab2180418d989d43 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-21T15:11:26Z |
| publishDate | 2020-04-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-bdac80ab8f06465aab2180418d989d432022-12-21T18:59:16ZengAIMS PressAIMS Mathematics2473-69882020-04-01542979299110.3934/math.2020192On the number of solutions of two-variable diagonal quartic equations over finite fieldsJunyong Zhao0Yang Zhao1Yujun Niu21 School of Mathematics and Statistics, Nanyang Institute of Technology, Nanyang 473004, P. R. China 2 Mathematical College, Sichuan University, Chengdu 610064, P. R. China3 Nanyang Normal University, Nanyang 473061, P. R. China1 School of Mathematics and Statistics, Nanyang Institute of Technology, Nanyang 473004, P. R. ChinaLet $p$ 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 $N(x_1^4+x_2^4=c)=q+O(q^{\frac{1}{2}}).$https://www.aimspress.com/article/10.3934/math.2020192/fulltext.htmldiagonal hypersurfacerational pointfinite fieldgauss sumjacobi sum |
| spellingShingle | Junyong Zhao Yang Zhao Yujun Niu On the number of solutions of two-variable diagonal quartic equations over finite fields AIMS Mathematics diagonal hypersurface rational point finite field gauss sum jacobi sum |
| title | On the number of solutions of two-variable diagonal quartic equations over finite fields |
| title_full | On the number of solutions of two-variable diagonal quartic equations over finite fields |
| title_fullStr | On the number of solutions of two-variable diagonal quartic equations over finite fields |
| title_full_unstemmed | On the number of solutions of two-variable diagonal quartic equations over finite fields |
| title_short | On the number of solutions of two-variable diagonal quartic equations over finite fields |
| title_sort | on the number of solutions of two variable diagonal quartic equations over finite fields |
| topic | diagonal hypersurface rational point finite field gauss sum jacobi sum |
| url | https://www.aimspress.com/article/10.3934/math.2020192/fulltext.html |
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