By: Francesco Amoroso (Contribution by), D. Bertrand (Contribution by), W.D. Brownawell (Contribution by), G. Diaz (Contribution by), M. Laurent (Contribution by)

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PrefaceList of Contributors Chapter 1. PHI (tau,z) and Transcendence1. Differential rings and modular forms2. Explicit differential equations3. Singular values4. Transcendence on phi and zChapter 2. Mahler's conjecture and other transcendence results1. Introduction2. A proof of Mahler's conjecture3. K. Barrè's work on modular functions4. Conjectures about modular and exponential functionsChapter 3. Algebraic independence for values of Ramanujan functions1. Main theorem and consequences2. How it can be proved?3. Constructions of the sequence of polynomials 4. Algebraic fundamentals5. Another proof of Theorem 1.1Chapter 4. Some remarks on proofs of algebraic independence1. Connection with elliptic functions2. Connection with modular series3. Another proof of algebraic independence of phi, ephi and TAU ( 1/4) 4. Approximation propertiesChapter 5. Elimination multihomogéne 1. Introduction2. Formes èliminantes des idèaux multihomogénes 3. Formes rèsultantes des idèaux multihomogénesChapter 6. Diophantine geometry 1. Elimination theory 2. Degree 3. Height 4. Geometric and arithmetic Bèzout theorems 5. Distance from a point to a variety 6. Auxiliary results 7. First metric Bèzout theorem 8. Second metric Bèzout theoremChapter 7. Gèomètrie diophantienne multiprojective 1. Introduction 2. Hauteurs 3. Une formule d'intersection 4. DistancesChapter 8. Criteria for algebraic independence 1. Criteria for algebraic independence 2. Mixed Segre- Veronese embeddings 3. Multi-projective criteria for algebraic independenceChapter 9. Upper bounds for (geometric ) Hilbert functions 1. The absolute case (following Kollàr) 2. The relative caseChapter 10. Multiplicity estimates for solutions of algebraic differential equations 1. Introduction 2. Reduction of Theorem 1.1 to bounds for polynomial ideals 3. Auxiliary assertions 4. End of the proof of Theorem 2.25. D-property for Ramanujan functionsChapter 11. Zero Estimates on Commutative Algebraic Groups 1. Introduction2. Degree of an intersection on an algebraic group3. Translation and derivations4. Statement and proof of the zero estimateChapter 12. Measures of algebraic independence for Mahler functions1. Theorems2. Proof of main theorem3. Proof of multiplicity estimateChapter 13. Algebraic Independence in Algebraic Groups. Part 1: Small Transcendence Degrees1. Introduction 2. General statements3. Concrete applications4. A criteria of algebraic independence with multiplicities 5. Introducing a matrix M6. The rank of the matrix M7. Analytic upper bound8. Proof of Proposition 5.1Chapter 14. Algebraic Independence in Algebraic Groups. Part 2: Large Transcendence Degrees1. Introduction2. Conjectures3. ProofsChapter 15. Some metric results in Transcendental Numbers Theory 1. Introduction2. One dimensional results3. Several dimensional results: "comparison Theorem"4. Several dimensional results: proof of Chudnovsky's conjectureChapter 16. The Hilbert Nullstellensatz, Inequalities for Polynomials, and Algebraic Independence1. The Hilbert Nullstellensatz and Effectivity 2. Liouville-Lojasiewicz Inequality3. The Lojasiewicz Inequality Implies the Nullstellensatz4. Geometric Version of the Nullstellensatz or Irrelevance of the Nullstellen Inequality for the Nullstellensatz5. Arithmetic Aspects of the Bèzout Version6. Some Algorithmic Aspects of the Bèzout VersionBibliographyIndex

Preface
List of Contributors
[Theta]([Gamma],[Zeta]) and Transcendence	p. 1
Mahler's conjecture and other transcendence results	p. 13
Algebraic independence for values of Ramanujan functions	p. 27
Some remarks on proofs of algebraic independence	p. 47
Elimination multihomogene	p. 53
Diophantine geometry	p. 83
Geometrie diophantienne multiprojective	p. 95
Criteria for algebraic independence	p. 133
Upper bounds for (geometric) Hilbert functions	p. 143
Multiplicity estimates for solutions of algebraic differential equations	p. 149
Zero Estimates on Commutative Algebraic Groups	p. 167
Measures of algebraic independence for Mahler functions	p. 187
Algebraic Independence in Algebraic Groups. Part I: Small Transcendence Degrees	p. 199
Algebraic Independence in Algebraic Groups. Part II: Large Transcendence Degrees	p. 213
Some metric results in Transcendental Numbers Theory	p. 227
The Hilbert Nullstellensatz, Inequalities for Polynomials, and Algebraic Independence	p. 239
Bibliography	p. 249
Index	p. 255
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540414964
ISBN-10: 3540414967
Series: Lecture Notes in Computer Science
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 276
Published: February 2001
Publisher: SPRINGER VERLAG GMBH
Dimensions (cm): 23.393 x 15.596  x 1.473
Weight (kg): 0.39