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 Independce 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 Expontial 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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