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Other titles in the Grundlehren Der Mathematischen Wissenschaften series:
Grundlehren Der Mathematischen Wissenschaften, #326: Diophantine Approximation on Linear Algebraic Groupsby Michael Waldschmidt
Synopses & Reviews
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.
Diophantine approximation is an important part of number theory, which studies integers and the relations between them. The theory of transcendental numbers is closely related to the study of Diophantine approximation. This book provides proofs for the main basic results dealing with transcendence properties, an introduction to height functions, and several proofs of Baker's theorem. The book also includes new results dealing with simultaneous approximation and of algebraic independence.
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.
Table of Contents
1. Introduction and Historical Survey Part I. Linear Independence of Logarithms of Algebraic Numbers 2. Transcendence Proofs in One Variable 3. Heights of Algebraic Numbers 4. The Criterion of Schneider-Lang 5. Zero Estimate 6. Linear Independence of Logarithms of Algebraic Numbers Part II. Measures of Linear Independence 7. A First Measure with a Simple Proof 8. Zero Estimate (Continued), by Damien ROY 9. Refined Measure III. Multiplicities in Higher Dimension 10. Multiplicity Estimates, by Damien ROY 11. Interpolation Determinants with One Derivative 12. On Baker's Method Part IV. The Linear Subgroup Theorem 13. Points Whose Coordinates are Logarithms of Algebraic Numbers 14. Lower Bounds for the Rank of Matrices Part V. Simultaneous Approximation of Values of the Exponential Function in Several Variables 15. A Quantitative Version of the Linear Subgroup Theorem 16. Applications to Diophantine Approximation 17. Algebraic Independence References
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