Approximation of complex algebraic numbers by algebraic numbers of bounded degree
by
Jan-Hendrik Evertse
Universiteit Leiden, Mathematisch Instituut, Postbus 9512, 2300 RA Leiden, The Netherlands
Coauthors: Yann Bugeaud (Strassbourg)

Define the height H(P) of P ∈ Z[X] to be the maximum of the absolute values of the coefficients of P. Further, for an algebraic number x, define its height H(x) to be the height of the minimal polynomial of x. For x ∈ C and for a positive integer n, denote by w_n(x) the supremum of all reals w with the property that there are infinitely many polynomials P ∈ Z[X] of degree at most n such that 0 < |P(x)| ≤ H(P)^-w. Further, define w_n^*(x) to be the supremum of all reals w^* such that there are infinitely algebraic numbers a ∈ C such that |x-a| ≤ H(a)^-w*-1. The functions w_n and w_n^* were introduced by Mahler and Koksma, respectively, and they play an important role in the classification of transcendental numbers.

It is known that w_n(x)=w_n^*(x)=min(n-1, d) for every real algebraic number x of degree d. This result is a consequence of W.M. Schmidt's celebrated Subspace Theorem from Diophantine approximation. As it turns out, the problem to compute w_n(x), w_n^*(x) for complex, non-real algebraic numbers x is more complicated. In my talk, I discuss some joint work with Yann Bugeaud in this direction, in which we determined w_n(x), w_n^*(x) for all complex, non-real algebraic numbers x of degree ≤ n+2 or ≥ 2n-1.

Date received: May 19, 2008