On Certain Polytopes Associated to Products of Algebraic Integer Conjugates

Let $d>k$ be positive integers. Motivated by an earlier result of Bugeaud and Nguyen, we let $E_{k,d}$ be the set of $(c_1,\dots ,c_k)\in {\mathbb{R}}_{\ge 0}^k$ such that $\mathrm{\mid }{\alpha }_0\mathrm{\mid }\mathrm{\mid }{\alpha }_1{\mathrm{\mid }}^{c_1}\cdots \mathrm{\mid }{\alpha }_k{\mathrm{\mid }}^{c_k}\ge 1$ for any algebraic integer $\alpha$ of degree $d$, where we label its Galois conjugates as ${\alpha }_0,\dots ,{\alpha }_{d-1}$ with $\mathrm{\mid }{\alpha }_0\mathrm{\mid }\ge \mathrm{\mid }{\alpha }_1\mathrm{\mid }\ge \cdots \ge \mathrm{\mid }{\alpha }_{d-1}\mathrm{\mid }$. 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 $(c_1,\dots ,c_k)\in E_{k,d}$ and for every $\alpha$ that is not a root of unity, the strict inequality $\mathrm{\mid }{\alpha }_0\mathrm{\mid }\mathrm{\mid }{\alpha }_1{\mathrm{\mid }}^{c_1}\cdots \mathrm{\mid }{\alpha }_k{\mathrm{\mid }}^{c_k}>1$ holds. We also provide a quantitative version of this inequality in terms of $d$ and the height of the minimal polynomial of $\alpha$.