65/3-4 (2004), 323–340 Distances between the conjugates of an algebraic number By JAN-HENDRIK EVERTSE (Leiden)

Abstract. Let K be a given number field of degree r> 3, denote by ξ 7 → ξ(i) (i = 1,..., r) the isomorphic embeddings of K into C, and let Σ be a subset of {1,..., r} of cardinality at least 2. Denote by M(α) the Mahler measure of an algebraic number α. By an elementary argument one shows that (∗) ∏{i,j}⊂Σ |α(i) − α(j) |> C · M(α)−κ holds for all α with K = Q(α), with C = 2−r(r−1)/2 and κ = r − 1. In the present paper we deduce inequalities (∗) with κ < r − 1 and with a constant C depending on K which are valid for all α with Q(α) = K. We obtain such inequalities with an ineffective constant C, using arguments and results from [7], [8], and with an effective constant C using a result from [9], [11]. Define κ(Σ) to be the infimum of all real numbers κ for which there exists a constant C> 0 such that (∗) holds for every α with Q(α) = K. Then clearly κ(Σ) 6 r − 1. We describe the sets Σ for which κ(Σ) = r − 1 and we give upper bounds for κ(Σ) in case that it is smaller than r − 1. For cubic fields we give the precise value of κ(Σ) for each set Σ. This solves a problem posed by Mignotte and Payafar [12, p. 187]