Introduction to Algebraic Independence Theory (Lecture Notes in Mathematics, 1752)

✍ Scribed by Yuri V. Nesterenko (editor), Patrice Philippon (editor)

• Publisher: Springer
• Year: 2001
• Tongue: English
• Leaves: 251
• Edition: 2001
• Category: Library

✦ Synopsis

In the last five years there has been very significant progress in the development of transcendence theory. A new approach to the arithmetic properties of values of modular forms and theta-functions was found. The solution of the Mahler-Manin problem on values of modular function j(tau) and algebraic independence of numbers pi and e^(pi) are most impressive results of this breakthrough. The book presents these and other results on algebraic independence of numbers and further, a detailed exposition of methods created in last the 25 years, during which commutative algebra and algebraic geometry exerted strong catalytic influence on the development of the subject.

✦ Table of Contents

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1752-front-matter(11s)
Titelei
Preface
Contents
1752-1(11s)
1. Differential rings and modular forms
2. Explicit differential equations
2.1. Eisenstein series
2.2. Modular theta functions
2.3. Modular functions
3. Singular values
3.1. Near holomorphy and modularity
3.2. Real analytic elliptic functions
3.3. Period relations
4. Transcendence on $tau$ and z
4.1. On the modular side
4.2. On the elliptic side
1752-13(14s)
1. Introduction
2. A proof of Mahler's conjecture
2.1. The modular invariant j and the discriminant
2.2. Modular polynomials (or modular "equations")
2.3. Measures of algebraic numbers and polynomials
2.4. Technical lemmas
2.5. A proof of Mahler's conjecture
2.6. An other proof of Mahler's conjecture
3. K. Barré's work on modular functions
3.1. Quantitative aspects of Mahler-Manin's conjecture
3.2. A modular proof of an old result of D.Bertrand
4. Conjectures about modular and exponential functions
4.1. Conjectures (C4E), (C4EW), (CDB)
4.2. Four weaker conjectures
1752-27(20s)
1. Main theorem and consequences.
1.1. Modular functions
1.2. Elliptic functions
1.3. Theta-functions
2. How it can be proved?
3. Construction of the sequence of polynomials
4. Algebraic fundamentals
5. Another proof of Theorem 1.1
1752-47(5s)
1. Connection with elliptic functions
2. Connection with modular series
3. Another proof of algebraic independence of $pi$, e'^$pi$ and $Gamma$(1/4)
4. Approximation properties
1752-53(29s)
1. Introduction
2. Formes éliminantes des idéaux multihomogènes
2.1. Idéaux éliminants
2.2. Polynome de Hilbert-Samuel multihomogène
2.3. Formes éliminantes
2.4. Spécialisation
2.5. Géométrie
3. Formes résultantes des idéaux multihomogènes
3.1. Forme associée à un module
3.2. Définition des formes résultantes
3.3. Degrés des formes résultantes
3.4. Spécialisation
3.5. Cas de I'idéal nul
1752-83(12s)
1. Elimination theory
2. Degree
3. Height
4. Geometric and arithmetic Bézout theorems
5. Distance from a point to a variety
6. Auxiliary results
7. First metric Bézout theorem
8. Second metric Bézout theorem
1752-95(37s)
1. Introduction
2. Hauteurs
2.1. Hauteurs des formes
2.2. Application aux formes résultantes
2.3. Application géométrique
2.4. Lien avec la géométrie d'Arakelov
3. Une formule d'intersection
3.1. Notations
3.2. Préparatifs
3.3. Résultat
4. Distances
4.1. Préliminaires
4.2. Définitions
4.3. Exemples
4.4. Décomposition de la distance
1752-133(9s)
1. Criteria for algebraic independence
2. Mixed Segre-Veronese embeddings
3. Multi-projective criteria for algebraic independence
1752-143(6s)
1. The absolute case (following Kollár)
1.1. Proof of (32)
1.2. Proof of (33)
2. The relative case
2.1. Proof of (34)
2.2. Proof of (35)
1752-149(17s)
1. Introduction
2. Reduction of Theorem 1.1 to bounds for polynomial ideals
3. Auxiliary assertions
4. End of the proof of Theorem 2.2
5. D-property for Rarnanujan functions
1752-167(19s)
1. Introduction
2. Degree of an intersection on an algebraic group
2.1. Algebraic groups
2.2. Degree of an ideal
2.3. Special ideals
3. Translations and derivations
3.1. Derivatives of functions along W
3.2. Operations on ideals
4. Statement and proof of the zero estimate
1752-187(11s)
1. Theorems
2. Proof of main theorem
3. Proof of multipicity estimate
1752-199(13s)
1. Introduction
2. General statements
3. Concrete applications
4. A criterion of algebraic independence with multiplicities
5. Introducing a matrix M
6. The rank of the matrix M
7. Analytic upper bound
8. Proof of Proposition 5.1
1752-213(13s)
1. Introduction
2. Conjectures
2.1. Commutative Algebraic Groups
2.2. Results: Large Transcendence Degree for the Values of the Exponential Function in One Variable
2.3. Historical Sketch
3. Proofs
3.1. Statement
3.2. Tools
3.2.1. Criterion of Algebraic Independence
3.2.2. Auxiliary Function
3.2.3. Zeros Estimate
1752-227(11s)
1. Introduction
2. One dimensional results
3. Several dimensional results: "comparison Theorem"
4. Several dimensional results: proof of Chudnovsky's conjecture
1752-239(10s)
1. The Hilbert Nullstellensatz and Effectivity
1.1. Classical (Ineffective) Versions
1.2. Relevance of Sharp Nullstellensatz for Independence
1.3. Application of Nullstellensatz for Algebraic Independence
2. Liouville-Lojasiewicz Inequality
2.1. The Inequality
2.2. Application of Inequality
2.3. Comparison with Philippon's Independence Criteria
3. The Lojasiewicz Inequality Implies the Nullstellensatz
4. Geometric Version of the Nullstellensatz or Irrelevance of the Nullstellen Inequality for the Nullstellensatz
4.1. Outline of Method of Proof
4.2. The Nullstellensatz Implies the Lojasiewicz Inequality
5. Arithmetic Aspects of the 134zout Version
6. Some Algorithmic Aspects of the B6zout Version
6.1. Nullstellensatz and Straight-Line Programs
6.2. Complexity Conjectures
6.3. Diophantine Questions
1752-back-matter(3s)
Index
List of Contributors