Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties o

Description: Diophantine Approximation on Linear Algebraic Groups by Michel Waldschmidt 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. FORMAT Paperback LANGUAGE English CONDITION Brand New Publisher Description 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. Notes Springer Book Archives 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 Lehmers problem, several proofs of Bakers theorem as well as explicit measures of linear independence of logarithms. An original feature is that proofs make systematic use of Laurents 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. 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 Bakers 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 authors 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 Promotional Springer Book Archives Long Description A transcendental number is a complex number which is not a root of a polynomial f E Z[X] \ {O}. Liouville constructed the first examples of transcendental numbers in 1844, Hermite proved the transcendence of e in 1873, Lindemann that of 1( in 1882. Siegel, and then Schneider, worked with elliptic curves and abelian varieties. After a suggestion of Cartier, Lang worked with commutative algebraic groups; this led to a strong development of the subject in connection with diophantine geometry, including Wiistholzs Analytic Subgroup Theorem and the proof by Masser and Wiistholz of Faltings Isogeny Theorem. In the meantime, Gelfond developed his method: after his solution of Hilberts seventh problem on the transcendence of afJ, he established a number of estimates from below for laf - a21 and lfillogal - loga21, where aI, a2 and fi are algebraic numbers. He deduced many consequences of such estimates for diophantine equations. This was the starting point of Bakers work on measures of linear independence oflogarithms of algebraic numbers. One of the most important features of transcendental methods is that they yield quantitative estimates related to algebraic numbers. This is one of the main reasons for which there are more mathematicians who deal with the transcendency of the special values of analytic functions than those who prove the algebraicity" I. A first example is Bakers method which provides lower bounds for nonvanishing numbers of the form lat! Review Quote "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 Feature Includes supplementary material: sn.pub/extras Details ISBN364208608X Author Michel Waldschmidt Year 2010 ISBN-10 364208608X ISBN-13 9783642086083 Format Paperback Publication Date 2010-12-06 Imprint Springer-Verlag Berlin and Heidelberg GmbH & Co. K Place of Publication Berlin Country of Publication Germany DEWEY 510 Birth 1956 Edition 1st Short Title DIOPHANTINE APPROXIMATION ON L Language English Media Book Series Number 326 Pages 633 Subtitle Transcendence Properties of the Exponential Function in Several Variables Illustrations XXIII, 633 p. Publisher Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Edition Description Softcover reprint of hardcover 1st ed. 2000 Series Grundlehren der mathematischen Wissenschaften Alternative 9783540667858 Audience Professional & Vocational We've got this At The Nile, if you're looking for it, we've got it. With fast shipping, low prices, friendly service and well over a million items - you're bound to find what you want, at a price you'll love! TheNile_Item_ID:96230937;

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