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El. knyga: Transcendental Number Theory

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, Foreword by (Universität Basel, Switzerland)
  • Formatas: PDF+DRM
  • Serija: Cambridge Mathematical Library
  • Išleidimo metai: 09-Jun-2022
  • Leidėjas: Cambridge University Press
  • Kalba: eng
  • ISBN-13: 9781009229968
Kitos knygos pagal šią temą:
Transcendental Number TheoryDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Cambridge Mathematical Library
  • Išleidimo metai: 09-Jun-2022
  • Leidėjas: Cambridge University Press
  • Kalba: eng
  • ISBN-13: 9781009229968
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First published in 1975, this classic book gives a systematic account of transcendental number theory, that is, the theory of those numbers that cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory, which continues to see rapid progress today. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalization of the ThueSiegelRoth theorem, of Shidlovsky's work on Siegel's E-functions and of Sprinduk's solution to the Mahler conjecture. This edition includes an introduction written by David Masser describing Baker's achievement, surveying the content of each chapter and explaining the main argument of Baker's method in broad strokes. A new afterword lists recent developments related to Baker's work.

Recenzijos

'Baker's book is the book on transcendental numbers. He covers a majority of those areas that have reached definitive results, presents most of the proofs in a complete yet far more compact form than hitherto available, and covers historical and bibliographical matters with great thoroughness and impeccable scholarship. As literature, it compares well with the finest works of Landau, Rademacher, and Titchmarsh.' Kenneth B. Stolarsky, Bulletin of the American Mathematical Society

Daugiau informacijos

Alan Baker's systematic account of transcendental number theory, with a new introduction and afterword explaining recent developments.
Introduction ix
David Masser
Preface xiii
1 The origins
1 Liouville's theorem
1(2)
2 Transcendence of e
3(3)
3 Lindemann's theorem
6(3)
2 Linear forms in logarithms
1 Introduction
9(2)
2 Corollaries
11(1)
3 Notation
12(1)
4 The auxiliary function
13(7)
5 Proof of main theorem
20(2)
3 Lower bounds for linear forms
1 Introduction
22(2)
2 Preliminaries
24(4)
3 The auxiliary function
28(6)
4 Proof of main theorem
34(2)
4 Diophantine equations
1 Introduction
36(2)
2 The Thue equation
38(2)
3 The hyperelliptic equation
40(3)
4 Curves of genus 1
43(1)
5 Quantitative bounds
44(3)
5 Class numbers of imaginary quadratic fields
1 Introduction
47(1)
2 L-functions
48(2)
3 Limit formula
50(1)
4 Class number 1
51(1)
5 Class number 2
52(3)
6 Elliptic functions
1 Introduction
55(1)
2 Corollaries
56(2)
3 Linear equations
58(1)
4 The auxiliary function
58(2)
5 Proof of main theorem
60(1)
6 Periods and quasi-periods
61(5)
7 Rational approximations to algebraic numbers
1 Introduction
66(3)
2 Wronskians
69(1)
3 The Index
69(4)
4 A combinatorial lemma
73(1)
5 Grids
74(1)
6 The auxiliary polynomial
75(1)
7 Successive minima
76(3)
8 Comparison of minima
79(2)
9 Exterior algebra
81(1)
10 Proof of main theorem
82(3)
8 Mahler's classification
1 Introduction
85(2)
2 A-numbers
87(1)
3 Algebraic dependence
88(1)
4 Heights of polynomials
89(1)
5 S-numbers
90(1)
6 U-numbers
90(2)
7 T-numbers
92(3)
9 Metrical theory
1 Introduction
95(1)
2 Zeros of polynomials
96(2)
3 Null sets
98(1)
4 Intersections of intervals
99(1)
5 Proof of main theorem
100(3)
10 The exponential function
1 Introduction
103(1)
2 Fundamental polynomials
104(4)
3 Proof of main theorem
108(1)
11 The Siegel-Shidlovsky theorems
1 Introduction
109(2)
2 Basic construction
111(3)
3 Further lemmas
114(1)
4 Proof of main theorem
115(3)
12 Algebraic independence
1 Introduction
118(2)
2 Exponential polynomials
120(2)
3 Heights
122(2)
4 Algebraic criterion
124(1)
5 Main arguments
125(4)
Bibliography 129(1)
Original papers 130(15)
Further publications 145(10)
New developments 155(7)
Some developments since 1990 by David Masser 162(4)
Index 166
Alan Baker was one of the leading British mathematicians of the past century. He took great strides in number theory by, among other achievements, obtaining a vast generalization of the GelfondSchneider Theorem and using it to give effective solutions to a large class of Diophantine problems. This work kicked off a new era in transcendental number theory and won Baker the Fields Medal in 1970. David Masser is Professor Emeritus in the Department of Mathematics and Computer Science at the University of Basel. He is a leading researcher in transcendence methods and applications and helped correct the proofs of the original edition of Transcendental Number Theory as Baker's student.
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