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 $(c_1,\ldots,c_k)\in\mathbb{R}_{\geq 0}^k$ 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 $α$.

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