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該書主要解普通指數函數e^z的值。一個關鍵的公開問題是超越數上的對數的代數無關性。
該書涵蓋了Hermite Lindemann定理、Gelfond-Schneider定理、6指數定理，通過探討萊默猜想介紹了高度函數, 貝克定理的證明和對數的線性獨立性的顯式測度。
該書的特色是系統地利用了勞倫特插值行列式來得出論據，最一般性的結論是所謂的線性群理論，新的是關於同時逼近和代數無關性的結論。
Michel Waldschmidt（M.瓦爾德施密特，法國）是國際知名學者，在數學界享有盛譽。本書凝聚了作者多年科研和教學成果，適用於科研工作者、高校教師和研究生。
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
