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| Main Author: | |
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| Format: | Preprint |
| Published: |
2016
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1608.04146 |
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| _version_ | 1866929743177187328 |
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| author | Chen, Evan |
| author_facet | Chen, Evan |
| contents | Let $k$ be a number field with cyclotomic closure $k^{\mathrm{cyc}}$, and let $h \in k^{\mathrm{cyc}}(x)$. For $A \ge 1$ a real number, we show that \[ \{ α\in k^{\mathrm{cyc}} : h(α) \in \overline{\mathbb Z} \text{ has house at most } A \} \] is finite for many $h$. We also show that for many such $h$ the same result holds if $h(α)$ is replaced by orbits $h(h(\cdots h(α)))$. This generalizes a result proved by Ostafe that concerns avoiding roots of unity, which is the case $A=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1608_04146 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Avoiding algebraic integers of bounded house in orbits of rational functions over cyclotomic closures Chen, Evan Number Theory 11R18, 37F10 Let $k$ be a number field with cyclotomic closure $k^{\mathrm{cyc}}$, and let $h \in k^{\mathrm{cyc}}(x)$. For $A \ge 1$ a real number, we show that \[ \{ α\in k^{\mathrm{cyc}} : h(α) \in \overline{\mathbb Z} \text{ has house at most } A \} \] is finite for many $h$. We also show that for many such $h$ the same result holds if $h(α)$ is replaced by orbits $h(h(\cdots h(α)))$. This generalizes a result proved by Ostafe that concerns avoiding roots of unity, which is the case $A=1$. |
| title | Avoiding algebraic integers of bounded house in orbits of rational functions over cyclotomic closures |
| topic | Number Theory 11R18, 37F10 |
| url | https://arxiv.org/abs/1608.04146 |