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 Ek,d be the set of (c₁,…,cₖ)≥ 0ᵏ such that α₀α₁c₁⋯αₖcₖ≥ 1 for any algebraic integer α of degree d, where we label its Galois conjugates as α₀,…,αd-1 with α₀≥ α₁≥⋯ ≥ αd-1. First, we give an explicit description of Ek,d as a polytope with 2ᵏ vertices. Then we prove that for $d>3k$, for every (c₁,…,cₖ)∈ Ek,d and for every α that is not a root of unity, the strict inequality α₀α₁c₁⋯αₖcₖ>1 holds. We also provide a quantitative version of this inequality in terms of d and the height of the minimal polynomial of α.