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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.13704 |
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| _version_ | 1866916133475450880 |
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| author | Young, Marley |
| author_facet | Young, Marley |
| contents | Given polynomials $f_1,\ldots,f_n$ in $m$ variables with integral coefficients, we give upper bounds for the number of integral $m$-tuples $\mathbf{u}_1,\ldots, \mathbf{u}_n$ of bounded height such that $f_1(\mathbf{u}_1), \ldots, f_n(\mathbf{u}_n)$ are multiplicatively dependent. We also prove, under certain conditions, a finiteness result for $\mathbf{u} \in \mathbb{Z}^m$ with relatively prime entries such that $f_1(\mathbf{u}),\ldots,f_n(\mathbf{u})$ are multiplicatively dependent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13704 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On multiplicatively dependent vectors of polynomial values Young, Marley Number Theory 11N25, 11C08, 11R04 Given polynomials $f_1,\ldots,f_n$ in $m$ variables with integral coefficients, we give upper bounds for the number of integral $m$-tuples $\mathbf{u}_1,\ldots, \mathbf{u}_n$ of bounded height such that $f_1(\mathbf{u}_1), \ldots, f_n(\mathbf{u}_n)$ are multiplicatively dependent. We also prove, under certain conditions, a finiteness result for $\mathbf{u} \in \mathbb{Z}^m$ with relatively prime entries such that $f_1(\mathbf{u}),\ldots,f_n(\mathbf{u})$ are multiplicatively dependent. |
| title | On multiplicatively dependent vectors of polynomial values |
| topic | Number Theory 11N25, 11C08, 11R04 |
| url | https://arxiv.org/abs/2402.13704 |