Infinite products with algebraic numbers
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910815211225088 |
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| author | Kristensen, Simon Laursen, Mathias Løkkegaard |
| author_facet | Kristensen, Simon Laursen, Mathias Løkkegaard |
| contents | We obtain general criteria for giving a lower bound on the degree of numbers of the form $\prod_{n=1}^\infty \left(1+\frac{b_n}{α_n}\right)$ or of the form $\prod_{m=1}^\infty \left(1+ \sum_{n=1}^\infty \frac{b_{n,m}}{α_{n,m}}\right)$, where the $α_n$ and $α_{n,m}$ are assumed to be algebraic integers, and the $b_n$ and $b_{n,m}$ are natural numbers. In each case, we give a lower bound of the degree over the smallest extension of $\mathbb{Q}$ containing all algebraic numbers in the expression. The criteria obtained depend on growth conditions on the involved quantities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03154 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinite products with algebraic numbers Kristensen, Simon Laursen, Mathias Løkkegaard Number Theory 11J72, 11J81 We obtain general criteria for giving a lower bound on the degree of numbers of the form $\prod_{n=1}^\infty \left(1+\frac{b_n}{α_n}\right)$ or of the form $\prod_{m=1}^\infty \left(1+ \sum_{n=1}^\infty \frac{b_{n,m}}{α_{n,m}}\right)$, where the $α_n$ and $α_{n,m}$ are assumed to be algebraic integers, and the $b_n$ and $b_{n,m}$ are natural numbers. In each case, we give a lower bound of the degree over the smallest extension of $\mathbb{Q}$ containing all algebraic numbers in the expression. The criteria obtained depend on growth conditions on the involved quantities. |
| title | Infinite products with algebraic numbers |
| topic | Number Theory 11J72, 11J81 |
| url | https://arxiv.org/abs/2502.03154 |