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Linear Independence of Logarithms of Algebraic Numbers

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

Abstract

In Chap. 4, we proved Baker’s homogeneous Theorem 1.5: if logarithms of algebraic numbers are linearly independent over ℚ, then they are linearly independent over \( \overline {\Bbb Q} \). The proof was an extension of Gel’fond’s solution to Hilbert’s seventh problem. Here we give a second proof of the same theorem, using an extension of Schneider’s method. The two main tools are an upper bound for the absolute value of an alternant in several variables (Proposition 6.6) and the zero estimate (namely Theorem 5.1).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

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

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