# Diophantine Approximation on Linear Algebraic Groups

## Transcendence Properties of the Exponential Function in Several Variables

Book

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

1. Front Matter
Pages I-XXIII
2. ### Introduction and Historical Survey

1. Michel Waldschmidt
Pages 1-27
3. ### Transcendence

1. Michel Waldschmidt
Pages 29-63
2. Michel Waldschmidt
Pages 65-114
3. Michel Waldschmidt
Pages 115-145
4. ### Linear Independence of Logarithms and Measures

1. Michel Waldschmidt
Pages 147-167
2. Michel Waldschmidt
Pages 169-186
3. Michel Waldschmidt
Pages 187-229
5. ### Multiplicities in Higher Dimension

1. Michel Waldschmidt
Pages 231-249
2. Michel Waldschmidt
Pages 251-316
3. Michel Waldschmidt
Pages 317-373
6. ### The Linear Subgroup Theorem

1. Michel Waldschmidt
Pages 375-416
2. Michel Waldschmidt
Pages 417-444
7. ### Simultaneous Approximation of Values of the Exponential Function in Several Variables

1. Michel Waldschmidt
Pages 445-494
2. Michel Waldschmidt
Pages 495-553
3. Michel Waldschmidt
Pages 555-614
8. Back Matter
Pages 615-636

### Introduction

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) on linear algebraic groups.

### Keywords

Algebra Diophantine approximation Exponential Functions Linear Algebraic groups Measures of Independence Simultaneous Diophantine Approximation Trancendental Numbers

#### Authors and affiliations

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

### 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
• Copyright Information Springer-Verlag Berlin Heidelberg 2000
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Hardcover ISBN 978-3-540-66785-8
• Softcover ISBN 978-3-642-08608-3
• eBook ISBN 978-3-662-11569-5
• Series ISSN 0072-7830
• Edition Number 1
• Number of Pages XXIII, 633
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site

## Reviews

"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