Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables: 326 (Grundlehren der mathematischen Wissenschaften, 326) - Softcover

Waldschmidt, Michel

Softcover

ISBN 10: 364208608X ISBN 13: 9783642086083

Publisher: Springer, 2010

Synopsis

The theory of transcendental numbers is closely related to the study of diophantine approximation. This book deals with values of the usual exponential function ez: a central open problem is the conjecture on algebraic independence of logarithms of algebraic numbers. Two chapters provide complete and simplified proofs of zero estimates (due to Philippon) on linear algebraic groups.

About this title

From the Back Cover

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.

Edition details

• Publisher: Springer
• Publication date: 2010
• Language: English
• ISBN 10: 364208608X
• ISBN 13: 9783642086083
• Binding: Paperback
• Number of pages: 656