The number of rational points on a class of hypersurfaces in quadratic extensions of finite fields
Let $ q $ be an even prime power and let $ \mathbb{F}_{q} $ be the finite field of $ q $ elements. Let $ f $ be a nonzero polynomial over $ \mathbb{F}_{q^2} $ of the form $ f = a_{1}x_{1}^{m_{1}}+\dots+a_{s}x_{s}^{m_{s}}+y_{1}y_{2}+\dots+y_{n-1}y_{n}+y_{n-2t-1}^{2}+\dots +y_{n-3}^2+y_{n-1}^{2}$ $+b_...
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AIMS Press
2023-06-01
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| Series: | Electronic Research Archive |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2023219?viewType=HTML |
| _version_ | 1827823146178183168 |
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| author | Qinlong Chen Wei Cao |
| author_facet | Qinlong Chen Wei Cao |
| author_sort | Qinlong Chen |
| collection | DOAJ |
| description | Let $ q $ be an even prime power and let $ \mathbb{F}_{q} $ be the finite field of $ q $ elements. Let $ f $ be a nonzero polynomial over $ \mathbb{F}_{q^2} $ of the form $ f = a_{1}x_{1}^{m_{1}}+\dots+a_{s}x_{s}^{m_{s}}+y_{1}y_{2}+\dots+y_{n-1}y_{n}+y_{n-2t-1}^{2}+\dots +y_{n-3}^2+y_{n-1}^{2}$ $+b_{t}y_{n-2t}^{2}+ $ $ \dots +b_{1}y_{n-2}^{2}+b_{0}y_{n}^{2} $, where $ a_{i}, b_j\in \mathbb{F}_{q^{2}}^{*}, $ $ m_i\ne 1, $ $ (m_{i}, m_{k}) = 1, $ $ i\ne k, $ $ m_{i}|(q+1), $ $ m_{i}\in \mathbb{Z}^{+}, $ $ 2|n $, $ n > 2 $, $ 0\leq t\leq \frac{n}{2}-2 $, $ {\mathrm{Tr}}_{\mathbb{F}_{q^2}/\mathbb{F}_{2}}(b_{j}) = 1 $ for $ i, k = 1, \dots, s $ and $ j = 0, 1, \dots, t $. For each $ b \in \mathbb{F}_{q^2} $, let $ N_{q^2}(f = b) $ denote the number of $ \mathbb{F}_{q^2} $-rational points on the affine hypersurface $ f = b $. In this paper, we obtain the formula of $ N_{q^2}(f = b) $ by using the Jacobi sums, Gauss sums and the results of quadratic form in finite fields. |
| first_indexed | 2024-03-12T02:07:01Z |
| format | Article |
| id | doaj.art-34c0c6bb7d2e47a09acb4694418e110e |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-03-12T02:07:01Z |
| publishDate | 2023-06-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj.art-34c0c6bb7d2e47a09acb4694418e110e2023-09-07T02:30:52ZengAIMS PressElectronic Research Archive2688-15942023-06-013174303431210.3934/era.2023219The number of rational points on a class of hypersurfaces in quadratic extensions of finite fieldsQinlong Chen 0Wei Cao 1School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, Fujian Province, ChinaSchool of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, Fujian Province, ChinaLet $ q $ be an even prime power and let $ \mathbb{F}_{q} $ be the finite field of $ q $ elements. Let $ f $ be a nonzero polynomial over $ \mathbb{F}_{q^2} $ of the form $ f = a_{1}x_{1}^{m_{1}}+\dots+a_{s}x_{s}^{m_{s}}+y_{1}y_{2}+\dots+y_{n-1}y_{n}+y_{n-2t-1}^{2}+\dots +y_{n-3}^2+y_{n-1}^{2}$ $+b_{t}y_{n-2t}^{2}+ $ $ \dots +b_{1}y_{n-2}^{2}+b_{0}y_{n}^{2} $, where $ a_{i}, b_j\in \mathbb{F}_{q^{2}}^{*}, $ $ m_i\ne 1, $ $ (m_{i}, m_{k}) = 1, $ $ i\ne k, $ $ m_{i}|(q+1), $ $ m_{i}\in \mathbb{Z}^{+}, $ $ 2|n $, $ n > 2 $, $ 0\leq t\leq \frac{n}{2}-2 $, $ {\mathrm{Tr}}_{\mathbb{F}_{q^2}/\mathbb{F}_{2}}(b_{j}) = 1 $ for $ i, k = 1, \dots, s $ and $ j = 0, 1, \dots, t $. For each $ b \in \mathbb{F}_{q^2} $, let $ N_{q^2}(f = b) $ denote the number of $ \mathbb{F}_{q^2} $-rational points on the affine hypersurface $ f = b $. In this paper, we obtain the formula of $ N_{q^2}(f = b) $ by using the Jacobi sums, Gauss sums and the results of quadratic form in finite fields.https://www.aimspress.com/article/doi/10.3934/era.2023219?viewType=HTMLfinite fieldpolynomialjacobi sumgauss sum |
| spellingShingle | Qinlong Chen Wei Cao The number of rational points on a class of hypersurfaces in quadratic extensions of finite fields Electronic Research Archive finite field polynomial jacobi sum gauss sum |
| title | The number of rational points on a class of hypersurfaces in quadratic extensions of finite fields |
| title_full | The number of rational points on a class of hypersurfaces in quadratic extensions of finite fields |
| title_fullStr | The number of rational points on a class of hypersurfaces in quadratic extensions of finite fields |
| title_full_unstemmed | The number of rational points on a class of hypersurfaces in quadratic extensions of finite fields |
| title_short | The number of rational points on a class of hypersurfaces in quadratic extensions of finite fields |
| title_sort | number of rational points on a class of hypersurfaces in quadratic extensions of finite fields |
| topic | finite field polynomial jacobi sum gauss sum |
| url | https://www.aimspress.com/article/doi/10.3934/era.2023219?viewType=HTML |
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