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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.00250 |
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| _version_ | 1866917738528636928 |
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| author | Albayrak, Seda Ghosh, Samprit Knapp, Greg Nguyen, Khoa D. |
| author_facet | Albayrak, Seda Ghosh, Samprit Knapp, Greg Nguyen, Khoa D. |
| contents | Let $d>k$ be positive integers. Motivated by an earlier result of Bugeaud and Nguyen, we let $E_{k,d}$ be the set of $(c_1,\ldots,c_k)\in\mathbb{R}_{\geq 0}^k$ such that $\vertα_0\vert\vertα_1\vert^{c_1}\cdots\vertα_k\vert^{c_k}\geq 1$ for any algebraic integer $α$ of degree $d$, where we label its Galois conjugates as $α_0,\ldots,α_{d-1}$ with $\vertα_0\vert\geq \vertα_1\vert\geq\cdots \geq \vertα_{d-1}\vert$. First, we give an explicit description of $E_{k,d}$ as a polytope with $2^k$ vertices. Then we prove that for $d>3k$, for every $(c_1,\ldots,c_k)\in E_{k,d}$ and for every $α$ that is not a root of unity, the strict inequality $\vertα_0\vert\vertα_1\vert^{c_1}\cdots\vertα_k\vert^{c_k}>1$ holds. We also provide a quantitative version of this inequality in terms of $d$ and the height of the minimal polynomial of $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00250 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Certain Polytopes Associated to Products of Algebraic Integer Conjugates Albayrak, Seda Ghosh, Samprit Knapp, Greg Nguyen, Khoa D. Number Theory Combinatorics 11J25 (Primary), 11C08 (Secondary) Let $d>k$ be positive integers. Motivated by an earlier result of Bugeaud and Nguyen, we let $E_{k,d}$ be the set of $(c_1,\ldots,c_k)\in\mathbb{R}_{\geq 0}^k$ such that $\vertα_0\vert\vertα_1\vert^{c_1}\cdots\vertα_k\vert^{c_k}\geq 1$ for any algebraic integer $α$ of degree $d$, where we label its Galois conjugates as $α_0,\ldots,α_{d-1}$ with $\vertα_0\vert\geq \vertα_1\vert\geq\cdots \geq \vertα_{d-1}\vert$. First, we give an explicit description of $E_{k,d}$ as a polytope with $2^k$ vertices. Then we prove that for $d>3k$, for every $(c_1,\ldots,c_k)\in E_{k,d}$ and for every $α$ that is not a root of unity, the strict inequality $\vertα_0\vert\vertα_1\vert^{c_1}\cdots\vertα_k\vert^{c_k}>1$ holds. We also provide a quantitative version of this inequality in terms of $d$ and the height of the minimal polynomial of $α$. |
| title | On Certain Polytopes Associated to Products of Algebraic Integer Conjugates |
| topic | Number Theory Combinatorics 11J25 (Primary), 11C08 (Secondary) |
| url | https://arxiv.org/abs/2408.00250 |