Explicit constructions of Diophantine tuples over finite fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917757121986560 |
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| author | Kim, Seoyoung Yip, Chi Hoi Yoo, Semin |
| author_facet | Kim, Seoyoung Yip, Chi Hoi Yoo, Semin |
| contents | A Diophantine $m$-tuple over a finite field $\mathbb{F}_q$ is a set $\{a_1,\ldots, a_m\}$ of $m$ distinct elements in $\mathbb{F}_{q}^{*}$ such that $a_{i}a_{j}+1$ is a square in $\mathbb{F}_q$ whenever $i\neq j$. In this paper, we study $M(q)$, the maximum size of a Diophantine tuple over $\mathbb{F}_q$, assuming the characteristic of $\mathbb{F}_q$ is fixed and $q \to \infty$. By explicit constructions, we improve the lower bound on $M(q)$. In particular, this improves a recent result of Dujella and Kazalicki by a multiplicative factor. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_05514 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit constructions of Diophantine tuples over finite fields Kim, Seoyoung Yip, Chi Hoi Yoo, Semin Number Theory 11D72, 11D45, 11T24, 11B83 A Diophantine $m$-tuple over a finite field $\mathbb{F}_q$ is a set $\{a_1,\ldots, a_m\}$ of $m$ distinct elements in $\mathbb{F}_{q}^{*}$ such that $a_{i}a_{j}+1$ is a square in $\mathbb{F}_q$ whenever $i\neq j$. In this paper, we study $M(q)$, the maximum size of a Diophantine tuple over $\mathbb{F}_q$, assuming the characteristic of $\mathbb{F}_q$ is fixed and $q \to \infty$. By explicit constructions, we improve the lower bound on $M(q)$. In particular, this improves a recent result of Dujella and Kazalicki by a multiplicative factor. |
| title | Explicit constructions of Diophantine tuples over finite fields |
| topic | Number Theory 11D72, 11D45, 11T24, 11B83 |
| url | https://arxiv.org/abs/2404.05514 |