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Diophantine approximation by conjugate algebraic integers

Published online by Cambridge University Press:  04 December 2007

Damien Roy
Département de Mathématiques, Université d'Ottawa, 585 King Edward, Ottawa, Ontario, K1N 6N5,
Michel Waldschmidt
Université Pierre et Marie Curie (Paris VI), Institut de Mathématiques CNRS UMR 7586, Théorie des Nombres, Case 247, 175 rue du Chevaleret, 75013 Paris,
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Building on the work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or p-adic number $\xi$ to be algebraic in terms of the existence of polynomials of bounded degree taking small values at $\xi$ together with most of their derivatives. The second one, which follows from this criterion by an argument of duality, is a result of simultaneous approximation by conjugate algebraic integers for a fixed number $\xi$ that is either transcendental or algebraic of sufficiently large degree. We also present several constructions showing that these results are essentially optimal.

Research Article
Foundation Compositio Mathematica 2004