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₊, ₃ be the set of (c₁, , cₖ) ₀ᵏ such that ₀₁^c₁ₖ^cₖ 1 for any algebraic integer of degree d, where we label its Galois conjugates as ₀, , ₃-₁ with ₀ ₁ ₃-₁. First, we give an explicit description of E₊, ₃ as a polytope with 2ᵏ vertices. Then we prove that for d>3k, for every (c₁, , cₖ) E₊, ₃ and for every that is not a root of unity, the strict inequality ₀₁^c₁ₖ^cₖ>1 holds. We also provide a quantitative version of this inequality in terms of d and the height of the minimal polynomial of.