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Grundlehren Der Mathematischen Wissenschaften Ser.: Diophantine Approximation on Linear Algebraic Groups : Transcendence Properties of the Exponential Function in Several Variables by Michel Waldschmidt (2000, Hardcover)
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About this product
Product Identifiers
PublisherSpringer Berlin / Heidelberg
ISBN-103540667857
ISBN-139783540667858
eBay Product ID (ePID)1709631
Product Key Features
Number of PagesXxiii, 633 Pages
Publication NameDiophantine Approximation on Linear Algebraic Groups : Transcendence Properties of the Exponential Function in Several Variables
LanguageEnglish
Publication Year2000
SubjectGroup Theory, Number Theory, Algebra / General, Geometry / Algebraic
TypeTextbook
AuthorMichel Waldschmidt
Subject AreaMathematics
SeriesGrundlehren Der Mathematischen Wissenschaften Ser.
FormatHardcover
Dimensions
Item Height0.6 in
Item Weight86.1 Oz
Item Length9.3 in
Item Width6.1 in
Additional Product Features
Intended AudienceScholarly & Professional
LCCN00-032967
Dewey Edition22
Reviews"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, "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
Series Volume Number326
Number of Volumes1 vol.
IllustratedYes
Dewey Decimal512.73
Table Of Content1. 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.
SynopsisThe 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., 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 Wiistholz's Analytic Subgroup Theorem and the proof by Masser and Wiistholz of Faltings' Isogeny Theorem. In the meantime, Gel'fond developed his method: after his solution of Hilbert's 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 Baker's 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 Baker's method which provides lower bounds for nonvanishing numbers of the form lat!·· .
LC Classification NumberQA150-272
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