Diophantine approximation by conjugate algebraic integers

Details Diophantine approximation by conjugate algebraic integers Diophantine approximation by conjugate algebraic integers

2002-07-12

arxiv.org/abs/math/0207102v2

Compositio Math. 140 (2004), 593--612.

Mathematics

Number Theory

The section 4 of this new version has been rewritten to simplify the proof of the main result. Other results in Sections 9 and

Scientific paper

10.1112/S0010437X03000708

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 $p$-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.

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