瓦尔德施密特所著的《线性代数群上的丢番图逼近(英文版)》主要解普通指数函数e^z的值，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。

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