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

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. It includes proofs of the main basic results (theorems of Hermite-Lindemann, Gelfond-Schneider, 6 exponentials theorem), an introduction to height functions and Lehmer's problem, several proofs of Baker's theorem as well as explicit measures of linear independence of logarithms. An original feature is the systematic use, in proofs, 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. Two chapters written by D. Roy provide complete and at the same time simplified proofs of zero estimates (due to Philippon) on linear algebraic groups.

Table of Contents:
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.

Review :
“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