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Diophantine Approximation on Linear Algebraic Groups

Transcendence Properties of the Exponential Function in Several Variables

  • Book
  • © 2000

Overview

Authors:
  1. Michel Waldschmidt

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

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

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Table of contents (15 chapters)

  1. Front Matter

    Pages I-XXIII
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  2. Introduction and Historical Survey

    1. Introduction and Historical Survey

      • Michel Waldschmidt
      Pages 1-27

  3. Transcendence

    1. Transcendence Proofs in One Variable

      • Michel Waldschmidt
      Pages 29-63

    2. Heights of Algebraic Numbers

      • Michel Waldschmidt
      Pages 65-114

    3. The Criterion of Schneider-Lang

      • Michel Waldschmidt
      Pages 115-145

  4. Linear Independence of Logarithms and Measures

    1. Zero Estimate, by Damien Roy

      • Michel Waldschmidt
      Pages 147-167

    2. Linear Independence of Logarithms of Algebraic Numbers

      • Michel Waldschmidt
      Pages 169-186

    3. Homogeneous Measures of Linear Independence

      • Michel Waldschmidt
      Pages 187-229

  5. Multiplicities in Higher Dimension

    1. Multiplicity Estimate by Damien Roy

      • Michel Waldschmidt
      Pages 231-249

    2. Refined Measures

      • Michel Waldschmidt
      Pages 251-316

    3. On Baker’s Method

      • Michel Waldschmidt
      Pages 317-373

  6. The Linear Subgroup Theorem

    1. Points Whose Coordinates are Logarithms of Algebraic Numbers

      • Michel Waldschmidt
      Pages 375-416

    2. Lower Bounds for the Rank of Matrices

      • Michel Waldschmidt
      Pages 417-444

  7. Simultaneous Approximation of Values of the Exponential Function in Several Variables

    1. A Quantitative Version of the Linear Subgroup Theorem

      • Michel Waldschmidt
      Pages 445-494

    2. Applications to Diophantine Approximation

      • Michel Waldschmidt
      Pages 495-553

    3. Algebraic Independence

      • Michel Waldschmidt
      Pages 555-614

  8. Back Matter

    Pages 615-636
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Keywords

About this book

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 algebraic numbers. This book includes proofs of the main basic results (theorems of Hermite-Lindemann, Gelfond-Schneider, 6 exponentials theorem), an introduction to height functions with a discussion of Lehmer's problem, several proofs of Baker's theorem as well as explicit measures of linear independence of logarithms. An original feature is that proofs make systematic use of Laurent's interpolation determinants. The most general result is the so-called Theorem of the Linear Subgroup, an effective version of which is also included. It yields new results of simultaneous approximation and of algebraic independence. 2 chapters written by D. Roy provide complete and at the same time simplified proofs of zero estimates (due to P. Philippon) onlinear algebraic groups.

Reviews

“This extensive monograph gives an excellent report on the present state of the art … . The reader having enough time and energy may learn from this carefully written book a great deal of modern transcendence theory from the very beginning. In this process, the many included exercises may be very helpful. Everybody interested in transcendence will certainly admire the author’s achievement to present such a clear and complete exposition of a topic growing so fast.” (P.Bundschuh, zbMATH 0944.11024, 2021)

"The present book is very nice to read, and gives a comprehensive overview of one wide aspect of Diophantine approximation. It includes the main achievements of the last several years, and points out the most interesting open questions. Moreover, each chapter is followed by numerous exercises, which provide an interesting complement of the main text. Many of them are adapted from original papers. Solutions are not given; however, there are helpful hints. This book is of great interest not only for experts in the field; it should also be recommended to anyone willing to have a taste of transcendental number theory. Undoubtedly, it will be very useful for anyone preparing a post-graduate course on Diophantine approximation."--MATHEMATICAL REVIEWS

Authors and Affiliations

  • Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie (Paris VI), Paris Cedex 05, France

    Michel Waldschmidt

Bibliographic Information

  • Book Title: Diophantine Approximation on Linear Algebraic Groups

  • Book Subtitle: Transcendence Properties of the Exponential Function in Several Variables

  • Authors: Michel Waldschmidt

  • Series Title: Grundlehren der mathematischen Wissenschaften

  • DOI: https://doi.org/10.1007/978-3-662-11569-5

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2000

  • Hardcover ISBN: 978-3-540-66785-8Published: 26 May 2000

  • Softcover ISBN: 978-3-642-08608-3Published: 06 December 2010

  • eBook ISBN: 978-3-662-11569-5Published: 14 March 2013

  • Series ISSN: 0072-7830

  • Series E-ISSN: 2196-9701

  • Edition Number: 1

  • Number of Pages: XXIII, 633

  • Topics: Algebra, Number Theory, Algebraic Geometry, Group Theory and Generalizations

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