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Transcendental Number Theory : Cambridge Mathematical Library - Alan Baker

Transcendental Number Theory

Cambridge Mathematical Library

Paperback Published: 3rd December 1990
ISBN: 9780521397919
Number Of Pages: 176

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First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalization of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's E-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979, however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.

Prefacep. ix
The origins
Liouville's theoremp. 1
Transcendence of ep. 3
Lindemann's theoremp. 6
Linear forms in logarithms
Introductionp. 9
Corollariesp. 11
Notationp. 12
The auxiliary functionp. 13
Proof of main theoremp. 20
Lower bounds for linear forms
Introductionp. 22
Preliminariesp. 24
The auxiliary functionp. 28
Proof of main theoremp. 34
Diophantine equations
Introductionp. 36
The Thue equationp. 38
The hyperelliptic equationp. 40
Curves of genus 1p. 43
Quantitative boundsp. 44
Class numbers of imaginary quadratic fields
Introductionp. 47
L-functionsp. 48
Limit formulap. 50
Class number 1p. 51
Class number 2p. 52
Elliptic functions
Introductionp. 55
Corollariesp. 56
Linear equationsp. 58
The auxiliary functionp. 58
Proof of main theoremp. 60
Periods and quasi-periodsp. 61
Rational approximations to algebraic numbers
Introductionp. 66
Wronskiansp. 69
The indexp. 69
A combinatorial lemmap. 73
Gridsp. 74
The auxiliary polynomialp. 75
Successive minimap. 76
Comparison of minimap. 79
Exterior algebrap. 81
Proof of main theoremp. 82
Mahler's classification
Introductionp. 85
A-numbersp. 87
Algebraic dependencep. 88
Heights of polynomialsp. 89
S-numbersp. 90
U-numbersp. 90
T-numbersp. 92
Metrical theory
Introductionp. 95
Zeros of polynomialsp. 96
Null setsp. 98
Intersections of intervalsp. 99
Proof of main theoremp. 100
The exponential function
Introductionp. 103
Fundamental polynomialsp. 104
Proof of main theoremp. 108
The Siegel-Shidlovsky theorems
Introductionp. 109
Basic constructionp. 111
Further lemmasp. 114
Proof of main theoremp. 115
Algebraic independence
Introductionp. 118
Exponential polynomialsp. 120
Heightsp. 122
Algebraic criterionp. 124
Main argumentsp. 125
Bibliographyp. 129
Original papersp. 130
Further publicationsp. 145
New developmentsp. 155
Indexp. 162
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521397919
ISBN-10: 052139791X
Series: Cambridge Mathematical Library
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 176
Published: 3rd December 1990
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 22.86 x 11.38  x 1.19
Weight (kg): 0.29

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