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

Abstract

Let n ≥ 1 be an integer. In the 1930's, Mahler and Koksma defined on the set C of complex numbers the functions w n and w * n , respectively, and used them to classify C into four classes. It turns out that both classifications are equivalent. However, when n ≥ 2, there exist complex numbers ξ for which w n (ξ) and w * n (ξ) are different. In the present note, we prove that the inequalities 0 ≤ w 2 (ξ) − w * 2 (ξ) ≤ 1 and 0 ≤ w 3 (ξ) − w * 3 (ξ) ≤ 2 are essentially best possible.

Distances between the conjugates of an algebraic number

“…Estimates for the distances between the conjugates of an algebraic number play an important role in complexity analyses of algorithms for polynomials. Further, they are of crucial importance in the study of the difference w n (ξ) − w * n (ξ), where w n (ξ), w * n (ξ) are quantities introduced by Mahler and Koksma, respectively, measuring how well a given transcendental complex number ξ can be approximated by algebraic numbers of degree n, see the three recent papers by Bugeaud [1], [2], [3].…”

“…Let K be a cubic field. Without loss of generality we assume that Σ = {1, 2}, where ξ → ξ (1) is real, ξ → ξ (2) is complex and ξ (3) = ξ (2) for ξ ∈ K.…”

“…We recall an argument of Mignotte and Payafar [12]. Let α with Q(α) = K. Then |α (1) − α (3) | = |α (1) − α (2) |, |α (2) − α (3) | |α (1) − α (2) | + |α (1)…”