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Catalogue Information
Field name	Details
Dewey Class	512
Title	Diophantine Approximation on Linear Algebraic Groups ([EBook] :) : Transcendence Properties of the Exponential Function in Several Variables / / by Michel Waldschmidt.
Author	Waldschmidt, Michel
Other name(s)	SpringerLink (Online service)
Publication	Berlin, Heidelberg : : Springer Berlin Heidelberg : : Imprint: Springer, , 2000.
Physical Details	XXIII, 633 p. : online resource.
Series	Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 0072-7830 ; ; 326
ISBN	9783662115695
Summary Note	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.:
Contents note	1. Introduction and Historical Survey -- 2. Transcendence Proofs in One Variable -- 3. Heights of Algebraic Numbers -- 4. The Criterion of Schneider-Lang -- 5. Zero Estimate, by Damien Roy -- 6. Linear Independence of Logarithms of Algebraic Numbers -- 7. Homogeneous Measures of Linear Independence -- 8. Multiplicity Estimates, by Damien Roy -- 9. Refined Measures -- 10. On Baker’s Method -- 11. Points Whose Coordinates are Logarithms of Algebraic Numbers -- 12. Lower Bounds for the Rank of Matrices -- 13. A Quantitative Version of the Linear Subgroup Theorem -- 14. Applications to Diophantine Approximation -- 15. Algebraic Independence -- References.
System details note	Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site	http://dx.doi.org/10.1007/978-3-662-11569-5
Links to Related Works	Subject References: Algebra . Algebraic Geometry . Group theory . Group Theory and Generalizations . Mathematics . Number Theory . Authors: Waldschmidt, Michel . Corporate Authors: SpringerLink (Online service) . Series: Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics . Classification: 512 .

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Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables /

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