Algebraic Independence

  • Michel Waldschmidt
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 326)

Abstract

Liouville’s inequality which has been used many times so far (namely Lemma 2.1) involves polynomial approximations: a tuple θ = (θ l,..., θ m ) of complex numbers for which there are polynomials f ∈ ℤ[X1,..., X m ] such that |f ( θ)| is small but not zero contains at least one transcendental element. The converse is also true, and this yields a transcendence criterion. A similar statement holds for algebraic approximations to a complex number: a number θ ∈ ℂ is transcendental if and only if there are algebraic numbers γ such that |θ − γ|is small but not zero. One deduces that numbers θ 1,..., θ m belonging to a field of transcendence degree 1 admit good simultaneous approximations by algebraic numbers γ l,..., γ m , where the quality of the approximation, namely the number max1≤ i m |θ i γ i |, is controlled in terms of the degree [ℚ(γ 1,..., γ m ): ℚ].

Keywords

Prool 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Michel Waldschmidt
    • 1
  1. 1.Institut de Mathématiques de JussieuUniversité Pierre et Marie Curie (Paris VI)Paris Cedex 05France

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