简介
该书主要解普通指数函数e^z的值。一个关键的公开问题是超越数上的对数的代数无关性。该书涵盖了Hermite Lindemann定理、Gelfond-Schneider定理、6指数定理，通过探讨莱默猜想介绍了高度函数, 贝克定理的证明和对数的线性独立性的显式测度。该书的特色是系统地利用了劳伦特插值行列式来得出论据，最一般性的结论是所谓的线性群理论，新的是关于同时逼近和代数无关性的结论。
目录
Prerequisites
Notation
1.Introduction and Historical Survey
1.1 Liouville.Hermite.Lindemann，Gel'fond，Baker
1.2 Lowef Bounds for|a1b1…ambm—1|
1.3 The Six Exponentials Theorem and the Four Exponentials Conjecture
1.4 Algebraic Independence of Logarithms
1.5 Diophantine Approximation on Linear Algebraic Groups Exercises
Part Ⅰ.Transcendence
2.Transcendence Proofs in One Variable
2.1 Inrroduction to Transcendence Proofs
2.2 Auxiliary Lemmas
2.3 Schneider's Method with Akemants—Real Case
2.4 Gel'fond's Method with Interpolation Determinants—Real Case
2.5 Gel'fond—Schneider's Theorem in the Complex Case
2.6 Hermite—Lindemann's Theorem in the Complex Case
Exercises
3.Heights of Algebraic Numbers
3.1 Absolute Values on a Numbef Field
3.2 The Absolute Logarithmic Height（Weil）
3.3 Mahler's Measure
3.4 Usual Height and Size
3.5 Liouville's Inequalities
3.6 Lower Bound for the Height
Open Problems
Exercises
Appendix—Inequalities Between Different Heights of a Polynomial—From a Manuscript by Alain Durand
4.The Criterion of Schneider Lang
4.1 Algebraic Values of Entifc Functions Satisfying Differenual Equauons
4.2 First Proof of Baker's Theorem
4.3 Schwarz' Lemma for Cartesian Products
4.4 Exponential Polynomials
4.5 Construction of an Auxiliary Function
4.6 Direct Proof of Corollary 4.2
Exercises
Part Ⅱ.Linear Independence of Logarithms and Measures
5.Zero Estimate，by Damien Roy
5.1 The Main Result
5.2 Some Algebraic Geomerry
5.3 The Group G and its Algebraic Subgroups
5.4 Proof of the Main Result
Exercises
6.Linear Independence of Logarithms of Algebraic Numbers
6.1 Applying the Zero Estimate
6.2 Upper Bounds for Altemants in Several Variables
6.3 A Second Proof of Baker's Homogeneous Theorem
Exercises
7.Homogeneous Measures of Linear Independence
7.1 Statement of the Measure
7.2 Lower Bound for a Zero Multiplicity
7.3 Upper Bound for the Arithmetic Determinant
7.4 Construction of a Nonzero Determinant
7.5 The Transcendence Argument—General Case
7.6 Proof of Theorem 7.1—General Case
7.7 The Rational Case： Fel'dman's Polynomials
7.8 Linear Dependence Relations between Logarithms
Open Problems
Exercises
Part Ⅲ.Multiplicities in Higher Dimension
8.Multiplicity Estimates，by Damien Roy
8.1 The Main Result
8.2 Some Commutative Algebra
8.3 The Group G and its Invariant Derivations
8.4 Proof of the Main Result
Exercises
9.Refined Measures
9.1 Second Proof of Baker's Nonhomogeneous Theorem
9.2 Proof of Theorem 9.1
9.3 Value of C（m）
9.4 Corollaries
Exercises
10.On Baker's Method
10.1 Linear Independence of Logarithms of Algebraic Numbers
10.2 Baker's Method with Interpolation Determinants
10.3 Baker's Method with Auxiliary Function
10.4 The State of the Art
Exercises
Part Ⅳ.The Linear Subgroup Theorem
11.Points Whose Coordinates are Logarithms of Algebraic Numbers
11.1 Introduction
11.2 One Parameter Subgroups
11.3 Six Variants of the Main Result
11.4 Linear Independcnce of Logarithms
11.5 Complex Toruses
11.6 Linear Combinations of Logarithms with Algebfaic Coefficients
11.7 Proof of the Linear Subgroup Theorem
Exercises
12.Lower Bounds for the Rank of Matrices
12.1 Entries are Linear Polynomials
12.2 Entries are Logarithms of Algebraic Numbers
12.3 Entries are Linear Combinations of Logarithms
12.4 Assuming the Conjecture on Algebraic Independence of Logarithms
12.5 Quadratic Relauons
Exercises
Part Ⅴ.Sunultaneous Apprmamation of Values of the Exponential Function in Several Variables
13.A Quantitative Version of the Linear Subgroup Theorem
13.1 The Main Result
13.2 Analytic Estimates
13.3 Exponcntial Polynomials
13.4 Proof of Theorem 13.1
13.5 Directions for Use
13.6 Introducing Feld' man's Polynomials
13.7 Duality： the Fouricr—Borel Transform
Exercises
14.Applications to Diophantine Approximation
14.1 A Quantitative Refinement to Gel'fond—Schneider's Theorem
14.2 A Quantitative Refinement to Hermite—Lindemann's Theorem
14.3 Simultaneous Approximation in Higher Dimension
14.4 Measures of Linear Independence of Logarithms（Again）
Open Problems
Exercises
15.Algebraic Independence
15.1 Criteria： Irrationality，Transcendence，Algebraic Independence
15.2 From Simultaneous Approximation to Algebraic Independence
15.3 Algcbraic Independence Results： Small Transcendence Degree
15.4 Large Transcendence Degree： Conjecture on Simultaneous
Approximation
15.5 Further Results and Conjectures
Exercises
References
Index
Notation
1.Introduction and Historical Survey
1.1 Liouville.Hermite.Lindemann，Gel'fond，Baker
1.2 Lowef Bounds for|a1b1…ambm—1|
1.3 The Six Exponentials Theorem and the Four Exponentials Conjecture
1.4 Algebraic Independence of Logarithms
1.5 Diophantine Approximation on Linear Algebraic Groups Exercises
Part Ⅰ.Transcendence
2.Transcendence Proofs in One Variable
2.1 Inrroduction to Transcendence Proofs
2.2 Auxiliary Lemmas
2.3 Schneider's Method with Akemants—Real Case
2.4 Gel'fond's Method with Interpolation Determinants—Real Case
2.5 Gel'fond—Schneider's Theorem in the Complex Case
2.6 Hermite—Lindemann's Theorem in the Complex Case
Exercises
3.Heights of Algebraic Numbers
3.1 Absolute Values on a Numbef Field
3.2 The Absolute Logarithmic Height（Weil）
3.3 Mahler's Measure
3.4 Usual Height and Size
3.5 Liouville's Inequalities
3.6 Lower Bound for the Height
Open Problems
Exercises
Appendix—Inequalities Between Different Heights of a Polynomial—From a Manuscript by Alain Durand
4.The Criterion of Schneider Lang
4.1 Algebraic Values of Entifc Functions Satisfying Differenual Equauons
4.2 First Proof of Baker's Theorem
4.3 Schwarz' Lemma for Cartesian Products
4.4 Exponential Polynomials
4.5 Construction of an Auxiliary Function
4.6 Direct Proof of Corollary 4.2
Exercises
Part Ⅱ.Linear Independence of Logarithms and Measures
5.Zero Estimate，by Damien Roy
5.1 The Main Result
5.2 Some Algebraic Geomerry
5.3 The Group G and its Algebraic Subgroups
5.4 Proof of the Main Result
Exercises
6.Linear Independence of Logarithms of Algebraic Numbers
6.1 Applying the Zero Estimate
6.2 Upper Bounds for Altemants in Several Variables
6.3 A Second Proof of Baker's Homogeneous Theorem
Exercises
7.Homogeneous Measures of Linear Independence
7.1 Statement of the Measure
7.2 Lower Bound for a Zero Multiplicity
7.3 Upper Bound for the Arithmetic Determinant
7.4 Construction of a Nonzero Determinant
7.5 The Transcendence Argument—General Case
7.6 Proof of Theorem 7.1—General Case
7.7 The Rational Case： Fel'dman's Polynomials
7.8 Linear Dependence Relations between Logarithms
Open Problems
Exercises
Part Ⅲ.Multiplicities in Higher Dimension
8.Multiplicity Estimates，by Damien Roy
8.1 The Main Result
8.2 Some Commutative Algebra
8.3 The Group G and its Invariant Derivations
8.4 Proof of the Main Result
Exercises
9.Refined Measures
9.1 Second Proof of Baker's Nonhomogeneous Theorem
9.2 Proof of Theorem 9.1
9.3 Value of C（m）
9.4 Corollaries
Exercises
10.On Baker's Method
10.1 Linear Independence of Logarithms of Algebraic Numbers
10.2 Baker's Method with Interpolation Determinants
10.3 Baker's Method with Auxiliary Function
10.4 The State of the Art
Exercises
Part Ⅳ.The Linear Subgroup Theorem
11.Points Whose Coordinates are Logarithms of Algebraic Numbers
11.1 Introduction
11.2 One Parameter Subgroups
11.3 Six Variants of the Main Result
11.4 Linear Independcnce of Logarithms
11.5 Complex Toruses
11.6 Linear Combinations of Logarithms with Algebfaic Coefficients
11.7 Proof of the Linear Subgroup Theorem
Exercises
12.Lower Bounds for the Rank of Matrices
12.1 Entries are Linear Polynomials
12.2 Entries are Logarithms of Algebraic Numbers
12.3 Entries are Linear Combinations of Logarithms
12.4 Assuming the Conjecture on Algebraic Independence of Logarithms
12.5 Quadratic Relauons
Exercises
Part Ⅴ.Sunultaneous Apprmamation of Values of the Exponential Function in Several Variables
13.A Quantitative Version of the Linear Subgroup Theorem
13.1 The Main Result
13.2 Analytic Estimates
13.3 Exponcntial Polynomials
13.4 Proof of Theorem 13.1
13.5 Directions for Use
13.6 Introducing Feld' man's Polynomials
13.7 Duality： the Fouricr—Borel Transform
Exercises
14.Applications to Diophantine Approximation
14.1 A Quantitative Refinement to Gel'fond—Schneider's Theorem
14.2 A Quantitative Refinement to Hermite—Lindemann's Theorem
14.3 Simultaneous Approximation in Higher Dimension
14.4 Measures of Linear Independence of Logarithms（Again）
Open Problems
Exercises
15.Algebraic Independence
15.1 Criteria： Irrationality，Transcendence，Algebraic Independence
15.2 From Simultaneous Approximation to Algebraic Independence
15.3 Algcbraic Independence Results： Small Transcendence Degree
15.4 Large Transcendence Degree： Conjecture on Simultaneous
Approximation
15.5 Further Results and Conjectures
Exercises
References
Index
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