ISBN: 3540414963
TITEL: Lecture Notes in Mathematics, Vol. 1752
AUTHOR: 
TOC:

Preface V
List of Contributors IX
Chapter 1. Theta (theta, z) and Transcendence 1
1. Differential rings and modular forms 1
2. Explicit differential equations 3
3. Singular values 5
4. Transcendence on theta and z 8
Chapter 2, Mahler's conjecture and other transcendence results 13
1. Introduction 13
2. A proof of Mahler's conjecture 14
3. K. Barr's work on modular functions 24
4. Conjectures about modular and exponential functions 25
Chapter 3. Algebraic independence for values of Ramanujan functions 27
1. Main theorem and consequences 27
2. How it can be proved? 32
3. Construction of the sequence of polynomials 33
4. Algebraic fundamentals 39
5. Another proof of Theorem 1.1. 43
Chapter 4. Some remarks on proofs of algebraic independence 47
1. Connection with elliptic functions 47
2. Connection with modular series 48
3. Another proof of algebraic independence of pi, e^pi and Gamma (1/4) 49
4. Approximation properties 50
Chapter 5. limination multihomogne 53
1. Introduction 53
2. Formes liminantes des idaux multihomognes 54
3. Formes rsultantes des idaux multihomogenes 71
Chapter 6. Diophantine geometry 83
1. Elimination theory 83
2. Degree 85
3. Height 85
4. Geometric and arithmetic B&out theorems 86
5. Distance from a point to a variety 88
6. Auxiliary results 90
7. First metric Bzout theorem 93
8. Second metric Bzout theorem 93
Chapter 7. Geometric diophantienne multiprojective 95
1. Introduction 95
2. Hauteurs 96
3. Une formule d'intersection 106
4. Distances 117
Chapter 8. Criteria for algebraic independence 133
1. Criteria for algebraic independence 136
2. Mixed Segre-Veronese embeddings 137
3. Multi-projective criteria for algebraic independence 140
Chapter 9. Upper bounds for (geometric) Hilbert functions 143
1. The absolute case (following Kollr) 144
2. The relative case 146
Chapter 10. Multiplicity estimates for soiutions of algebraic differential equations 149
1. Introduction 149
2. Reduction of Theorem 1.1 to bounds for polynomial ideals. 153
3. Auxiliary assertions. 154
4. End of the proof of Theorem 2.2. 159
5. D-property for Ramanujan functions. 161
Chapter 11. Zero Estimates on Commutative Algebraic Groups 167
1. Introduction 167
2. Degree of an intersection on an algebraic group 167
3. Translations and derivations 173
4. Statement and proof of the zero estimate 181
Chapter 12. Measures of algebraic independence for Mahler functions 187
1. Theorems 187
2. Proof of main theorem 188
3. Proof of multipicity estimate 194
Chapter 13. Algebraic Independence in Algebraic Groups.
Part I: Small Transcendence Degrees 199
1. Introduction 199
2. General statements. 199
3. Concrete applications. 201
4. A criterion of algebraic independence with multiplicities. 203
5. Introducing a matrix M. 205
6. The rank of the matrix M. 206
7. Analytic upper bound. 208
8. Proof of Proposition 5.1. 210
Chapter 14. Algebraic Independence in Algebraic Groups.
Part II: Large Transcendence Degrees 213
1. Introduction 213
2. Conjectures 213
3. Proofs 217
Chapter 15. Some metric results in Teanscendental Numbers Theory 227
1. Introduction 227
2. One dimensional results 228
3. Several dimensional results: "comparison Theorem" 230
4. Several dimensional results: proof of Chudnovsky's conjecture 236
Chapter 16. The Hilbert Nullstellensatz, Inequalities for Polynomials, and Algebraic Independence 239
1. The Hilbert Nu~steilensatz and Effectivity 239
2. Liouville-Lojasiewicz Inequality 242
3. The Lojasiewicz Inequality Implies the Nullstellensatz 244
4. Geometric Version of the Nullstellensatz or Irrelevance of the Nullstelfen Inequality for the Nullstellensatz 245
5. Arithmetic Aspects of the Bzout Version 246
6. Some Algorithmic Aspects of the Bzout Version 247
Bibliography 249
Index 255
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