On Certain Polytopes Associated to Products of Algebraic Integer Conjugates

Abstract

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 $|\hat{I}{\pm }_0||\hat{I}{\pm }_1{|}^{c_1}\cdots |\hat{I}{\pm }_k{|}^{c_k}\ge 1$ for any algebraic integer $\hat{I}\pm$ of degree $d$, where we label its Galois conjugates as $\hat{I}{\pm }_0,\dots ,\hat{I}{\pm }_{d-1}$ with $|\hat{I}{\pm }_0|\ge |\hat{I}{\pm }_1|\ge \cdots \ge |\hat{I}{\pm }_{d-1}|$. 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 $\hat{I}\pm$ that is not a root of unity, the strict inequality $|\hat{I}{\pm }_0||\hat{I}{\pm }_1{|}^{c_1}\cdots |\hat{I}{\pm }_k{|}^{c_k}>1$ holds. We also provide a quantitative version of this inequality in terms of $d$ and the height of the minimal polynomial of $\hat{I}\pm$.

15 pages