Diophantine approximation by conjugate algebraic integers
摘要
Building on 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 -adic number to be algebraic in terms of the existence of polynomials of bounded degree taking small values at 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 that is either transcendental or algebraic of sufficiently large degree. We also present several constructions showing that these results are essentially optimal.
引用
@article{arxiv.math/0207102,
title = {Diophantine approximation by conjugate algebraic integers},
author = {Damien Roy and Michel Waldschmidt},
journal= {arXiv preprint arXiv:math/0207102},
year = {2007}
}
备注
The section 4 of this new version has been rewritten to simplify the proof of the main result. Other results in Sections 9 and 10 have been improved. To appear in Compositio Math