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Algebraic Independence is an expanded version of the notes of a course of lectures given by Professor Yuri V. Nesterenko at TIFR. It deals with several important results and methods in Transcendental Number Theory. The main results which are proved in detail are the classical result of Lindemann-Weierstrass, Siegel?s theory of E-functions and Shidlovskii?s theorem on the algebraic independence of the values of the E-functions, the Gelfond-Schneider Theorem using interpolation determinants and the famous result of the author in 1996 on the algebraic independence of the values of the Ramanujan functions. The book is self-contained and the proofs are clear and lucid. Brief history of the topics is also given. Table of Contents Preface / Lindemann-Weierstrass Theorem / E-functions and Shidlovskii?s Theorem / Small Transcendence Degree (Exponential Function) / Small Transcendence Degree (Modular Functions) / Algebraic Fundamentals / Philippon?s Criterion of Algebraic Independence / Fields of Large Transcendence Degree / Multiplicity Estimates / Bibliography / Index. Printed Pages: 160. Buchnummer des Verkäufers 65990
Inhaltsangabe: This book is an expanded version of the notes of a course of lectures given by at the Tata Institute of Fundamental Research in 1998. It deals with several important results and methods in transcendental number theory. First, the classical result of Lindemann-Weierstrass and its applications are dealt with. Subsequently, Siegel's theory of E-functions is developed systematically, culminating in Shidlovskii's theorem on the algebraic independence of the values of the E-functions satisfying a system of differential equations at certain algebraic values. Proof of the Gelfond-Schneider Theorem is given based on the method of interpolation determinants introduced in 1992 by M. Laurent.The author's famous result in 1996 on the algebraic independence of the values of the Ramanujan functions is the main theme of the reminder of the book. After deriving several beautiful consequences of his result, the author develops the algebraic material necessary for the proof. The two important technical tools in the proof are Philippon's criterion for algebraic independence and zero bound for Ramanujan functions. The proofs of these are covered in detail.The author also presents a direct method, without using any criterion for algebraic independence as that of Philippon, by which one can obtain lower bounds for transcendence degree of finitely generated field Q([omega]1,. ..,[omega]m). This is a contribution towards Schanuel's conjecture.The book is self-contained and the proofs are clear and lucid. A brief history of the topics is also given. Some sections intersect with Chapters 3 and 10 of "Introduction to Algebraic Independence Theory, Lecture Notes in Mathematics, Springer, 1752", edited by Yu. V. Nesterenko and P. Philippon.
About the Author: Yu. V. Nesterenko.: Faculty of Mechanics and Mathematics Moscow State University, 119899 Moscow (Russia)
Titel: Algebraic Independence
Verlag: Narosa Publishing House
Auflage: 5th or later edition.
Buchbeschreibung Suomalainen Tiedeakatemia, Helsinki, 1972. original kartoniert, gr.-8°, 45 S.; Zustand: gut, Klammerheftung angerostet an 150 Buch. Artikel-Nr. JB2-267
Buchbeschreibung Artikel-Nr. 7503