تقرير
On Certain Polytopes Associated to Products of Algebraic Integer Conjugates
| العنوان: | On Certain Polytopes Associated to Products of Algebraic Integer Conjugates |
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| المؤلفون: | Albayrak, Seda, Ghosh, Samprit, Knapp, Greg, Nguyen, Khoa D. |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Number Theory, Mathematics - Combinatorics, 11J25 (Primary), 11C08 (Secondary) |
| الوصف: | 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 $\vert\alpha_0\vert\vert\alpha_1\vert^{c_1}\cdots\vert\alpha_k\vert^{c_k}\geq 1$ for any algebraic integer $\alpha$ of degree $d$, where we label its Galois conjugates as $\alpha_0,\ldots,\alpha_{d-1}$ with $\vert\alpha_0\vert\geq \vert\alpha_1\vert\geq\cdots \geq \vert\alpha_{d-1}\vert$. 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 $\alpha$ that is not a root of unity, the strict inequality $\vert\alpha_0\vert\vert\alpha_1\vert^{c_1}\cdots\vert\alpha_k\vert^{c_k}>1$ holds. We also provide a quantitative version of this inequality in terms of $d$ and the height of the minimal polynomial of $\alpha$. Comment: 15 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2408.00250 |
| رقم الأكسشن: | edsarx.2408.00250 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |