标题： 关于与代数整数共轭乘积相关的某些多面体 显示英文标题

标题： On Certain Polytopes Associated to Products of Algebraic Integer Conjugates

摘要： 设$d>k$为正整数。 受Bugeaud和Nguyen先前结果的启发，我们令$E_{k,d}$为满足以下条件的$(c_1,\ldots,c_k)\in\mathbb{R}_{\geq 0}^k$的集合：对于任何次数为$d$的代数整数$\alpha$，$\vert\alpha_0\vert\vert\alpha_1\vert^{c_1}\cdots\vert\alpha_k\vert^{c_k}\geq 1$成立，其中我们将它的伽罗瓦共轭标记为$\alpha_0,\ldots,\alpha_{d-1}$，其中$\vert\alpha_0\vert\geq \vert\alpha_1\vert\geq\cdots \geq \vert\alpha_{d-1}\vert$。 首先，我们给出$E_{k,d}$作为具有$2^k$个顶点的多面体的显式描述。 然后我们证明对于$d>3k$，对于每个$(c_1,\ldots,c_k)\in E_{k,d}$以及对于每个不是单位根的$\alpha$，严格不等式$\vert\alpha_0\vert\vert\alpha_1\vert^{c_1}\cdots\vert\alpha_k\vert^{c_k}>1$成立。 我们还提供了该不等式的定量版本，涉及$d$和$\alpha$的最小多项式的高度。 显示英文摘要

摘要： 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 $\alpha$.