Diophantine Approximation on Linear Algebraic Groups Transcendence P
The theory of transcendental numbers is closely related to the study of diophantine approximation. This book deals with values of the usual exponential function e^z. A central open problem is the conjecture on algebraic independence of logarithms of algeb
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Editors S. S. Chern B. Eckmann P. de la Harpe H. Hironaka F. Hirzebruch N. Hitchin L. Hormander M.-A. Knus A. Kupiainen J. Lannes G. Lebeau M. Ratner D. Serre Ya. G. Sinai N. J. A. Sloane J. Tits M. Waldschmidt S. Watanabe
Managing Editors M. Berger
J. Coates S. R. S. Varadhan
Springer-Verlag Berlin Heidelberg GmbH
Michel Waldschmidt
Diophantine Approximation on Linear Algebraic Groups Transcendence Properties of the Exponential Function in Several Variables
Springer
Michel Waldschmidt Institut de Mathematiques de Jussieu Universite Pierre et Marie Curie (Paris VI) Case 247 75252 Paris Cedex 05, France e-mail: [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Waldschmidt. Michel: Diophantine approximation on linear algebraic groups: transcendence properties of the exponential function in several variables / Michel Waldschmidt. (Grundlehren der mathematischen Wissenschailen ; 326) ISBN 978-3-642-08608-3 ISBN 978-3-662-11569-5 (eBook) DOI 10.1007/978-3-662-11569-5
Mathematics Subject Classification (1991): 11-02, nIxx, 14Lxx, 20Gxx, 33BlO ISSN 0072-7830 ISBN 978-3-642-08608-3
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Originally published by Springer-Verlag Berlin Heidelberg New York in 2000 Softcover reprint of the hardcover 1St edition 2000 Cover design: MetaDesign plus GmbH, Berlin Typeset by the author using the MathTime fonts. Printed on acid-free paper SPIN: 10691667
41/3143LK-5 43210
Preface
A transcendental number is a complex number which is not a root of a polynomial f E Z[X] \ {O}. Liouville constructed the first examples of transcendental numbers in 1844, Hermite proved the transcendence of e in 1873, Lindemann that of 1'( in 1882. Siegel, and then Schneider, worked with elliptic curves and abelian varieties. After a suggestion of Cartier, Lang worked with commutative algebraic groups; this led to a strong development of the subject in connection with diophantine geometry, including Wiistholz's Analytic Subgroup Theorem and the proof by Masser and Wiistholz of Faltings' Isogeny Theorem. In the meantime, Gel'fond developed his method: after his solution of Hilbert's seventh problem on the transcendence of afJ, he established a number of estimates from below for laf - a21 and lfillogal - loga21, where aI, a2 and fi are algebraic numbers. He deduced many consequences of such estimates for diophantine equations. This was the startin
