Distances between the conjugates of an algebraic number

Abstract

Let K be a given number field of degree r 3, denote by ξ → 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) 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].

Mahler's classification of numbers compared with Koksma's. III.

“…It is wellknown (see e.g. [5]) that there exists a positive constant c(n), depending only on n, such that for any subset Σ of {1, . .…”

“…This estimate is best possible when |Σ| = n (see e.g. Theorem 1.2 from [5]) and for |Σ| = 2 and n = 3 (see [8]). Namely, let (p m /q m ) m≥1 denote the sequence of convergents to √ 2.…”

“…We point out that the roots of the polynomials from the above families are uniformly bounded (unlike in the construction given in [5]).…”

“…Our proof rests on the construction of families of quadratic and cubic irreducible, integer polynomials having two real roots very close to each other. It makes use of a recent result of Evertse [5].…”

Polynomials and binary forms with given discriminant

“…Our Corollary 5 has been recently applied by Evertse [13] to distances between the conjugates of an algebraic number. We note that in [29], [32], [34] and [16] there are other consequences and applications as well of the earlier versions of our Theorems 1, 2 and 3, e.g.…”