ISBN: 3-540-66785-7
TITLE: Diophantine Approximation on Linear Algebraic Groups
AUTHOR: Waldschmidt, Michel
TOC:

Prerequisites XIII
Notation XIX
1. Introduction and Historical Survey 1
1.1 Liouville, Hermite, Lindemann, Gel'fond, Baker 1
1.2 Lower Bounds for |a_1^b1... a_^bm - 1| 6
1.3 The Six Exponentials Theorem and the Four Exponentials Conjecture 13
1.4 Algebraic Independence of Logarithms 15
1.5 Diophantine Approximation on Linear Algebraic Groups 19
Exercises 21
Part I. Transcendence
2. Transcendence Proofs in One Variable 29
2.1 Introduction to Transcendence Proofs 29
2.2 Auxiliary Lemmas 33
2.3 Schneider's Method with Alternants  Real Case 37
2.4 Gel'fond's Method with Interpolation Determinants  Real Case 43
2.5 Gel'fond-Schneider's Theorem in the Complex Case 49
2.6 Hermite-Lindemann's Theorem in the Complex Case 55
Exercises 59
3. Heights of Algebraic Numbers 65
3.1 Absolute Values on a Number Field 66
3.2 The Absolute Logarithmic Height (Weil) 75
3.3 Mahler's Measure 78
3.4 Usual Height and Size 80
3.5 Liouville's Inequalities 82
3.6 Lower Bound for the Height 86
Open Problems 105
Exercises 106
Appendix  Inequalities Between Different Heights of a Polynomial  From a Manuscript by Alain Durand 113
4. The Criterion of Schneider-Lang 115
4.1 Algebraic Values of Entire Functions Satisfying Differential Equations 115
4.2 First Proof of Baker's Theorem 118
4.3 Schwarz' Lemma for Cartesian Products 122
4.4 Exponential Polynomials 130
4.5 Construction of an Auxiliary Function 131
4.6 Direct Proof of Corollary 4.2 136
Exercises 141
Part II. Linear Independence of Logarithms and Measures
5. Zero Estimate, by Damien Roy 147
5.1 The Main Result 147
5.2 Some Algebraic Geometry 150
5.3 The Group G and its Algebraic Subgroups 156
5.4 Proof of the Main Result 164
Exercises 166
6. Linear Independence of Logarithms of Algebraic Numbers 169
6.1 Applying the Zero Estimate 170
6.2 Upper Bounds for Alternants in Several Variables 175
6.3 A Second Proof of Baker's Homogeneous Theorem 181
Exercises 184
7. Homogeneous Measures of Linear Independence 187
7.1 Statement of the Measure 187
7.2 Lower Bound for a Zero Multiplicity 192
7.3 Upper Bound for the Arithmetic Determinant 195
7.4 Construction of a Nonzero Determinant 199
7.5 The Transcendence Argument - General Case 203
7.6 Proof of Theorem 7.1 - General Case 208
7.7 The Rational Case: Fel'dman's Polynomials 214
7.8 Linear Dependence Relations between Logarithms 222
Open Problems 227
Exercises 227
Part III. Multiplicities in Higher Dimension
8. Multiplicity Estimates, by Damien Roy 231
8.1 The Main Result 231
8.2 Some Commutative Algebra 234
8.3 The Group G and its Invariant Derivations 238
8.4 Proof of the Main Result 245
Exercises 247
9. Refined Measures 251
9.1 Second Proof of Baker's Nonhomogeneous Theorem 252
9.2 Proof of Theorem 9.1 262
9.3 Value of C(m) 286
9.4 Corollaries 302
Exercises 314
10. On Baker's Method 317
10.1 Linear Independence of Logarithms of Algebraic Numbers 317
10.2 Baker's Method with Interpolation Determinants 329
10.3 Baker's Method with Auxiliary Function 356
10.4 The State of the Art 360
Exercises 371
Part IV. The Linear Subgroup Theorem
11. Points Whose Coordinates are Logarithms of Algebraic Numbers 375
11.1 Introduction 375
11.2 One Parameter Subgroups 379
11.3 Six Variants of the Main Result 381
11.4 Linear Independence of Logarithms 387
11.5 Complex Toruses 394
11.6 Linear Combinations of Logarithms with Algebraic Coefficients 398
11.7 Proof of the Linear Subgroup Theorem 404
Exercises 411
12. Lower Bounds for the Rank of Matrices 417
12.1 Entries are Linear Polynomials 418
12.2 Entries are Logarithms of Algebraic Numbers 432
12.3 Entries are Linear Combinations of Logarithms 435
12.4 Assuming the Conjecture on Algebraic Independence of Logarithms 437
12.5 Quadratic Relations 438 
Exercises 441
Part V. Simultaneous Approximation of Values of the Exponential Function in Several Variables
13. A Quantitative Version of the Linear Subgroup Theorem 445
13.1 The Main Result 447
13.2 Analytic Estimates 450
13.3 Exponential Polynomials 459
13.4 Proof of Theorem 13.1 464
13.5 Directions for Use 471
13.6 Introducing Feld'man's Polynomials 476
13.7 Duality: the Fourier-Borel Transform 480
Exercises 490
14. Applications to Diophantine Approximation 495
14.1 A Quantitative Refinement to Gel'fond-Schneider's Theorem 496
14.2 A Quantitative Refinement to Hermite-Lindemann's Theorem 510
14.3 Simultaneous Approximation in Higher Dimension 521
14.4 Measures of Linear Independence of Logarithms (Again) 536
Open Problems 547
Exercises 549
15. Algebraic Independence 555
15.1 Criteria: Irrationality, Transcendence, Algebraic Independence 555
15.2 From Simultaneous Approximation to Algebraic Independence 569
15.3 Algebraic Independence Results: Small Transcendence Degree 587
15.4 Large Transcendence Degree: Conjecture on Simultaneous Approximation 594
15.5 Further Results and Conjectures 598
Exercises 606
References 615
Index 629
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