Introduction to Algebraic Independence Theory

ISBN-13: 9783540414964 / Angielski / Miękka / 2001 / 260 str.

Y. V. Nesterenko; P. Philippon; Yuri V. Nesterenko

Opis

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 DEGREES(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 s

Spis treści

Preface List of Contributors Chapter 1. PHI (tau,z) and Transcendence 1. Differential rings and modular forms 2. Explicit differential equations 3. Singular values 4. Transcendence on phi and z Chapter 2. Mahler's conjecture and other transcendence results 1. Introduction 2. A proof of Mahler's conjecture 3. K. Barré's work on modular functions 4. Conjectures about modular and exponential functions Chapter 3. Algebraic independence for values of Ramanujan functions 1. Main theorem and consequences 2. How it can be proved? 3. Constructions of the sequence of polynomials 4. Algebraic fundamentals 5. Another proof of Theorem 1.1 Chapter 4. Some remarks on proofs of algebraic independence 1. Connection with elliptic functions 2. Connection with modular series 3. Another proof of algebraic independence of phi, ephi and TAU ( 1/4) 4. Approximation properties Chapter 5. Élimination multihomogène 1. Introduction 2. Formes éliminantes des idéaux multihomogènes 3. Formes résultantes des idéaux multihomogènes Chapter 6. Diophantine geometry 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 Chapter 7. Géométrie diophantienne multiprojective 1. Introduction 2. Hauteurs 3. Une formule d'intersection 4. Distances Chapter 8. Criteria for algebraic independence 1. Criteria for algebraic independence 2. Mixed Segre- Veronese embeddings 3. Multi-projective criteria for algebraic independence Chapter 9. Upper bounds for (geometric ) Hilbert functions 1. The absolute case (following Kollár) 2. The relative case Chapter 10. Multiplicity estimates for solutions of algebraic differential equations 1. Introduction 2. Reduction of Theorem 1.1 tobounds for polynomial ideals 3. Auxiliary assertions 4. End of the proof of Theorem 2.2 5. D-property for Ramanujan functions Chapter 11. Zero Estimates on Commutative Algebraic Groups 1. Introduction 2. Degree of an intersection on an algebraic group 3. Translation and derivations 4. Statement and proof of the zero estimate Chapter 12. Measures of algebraic independence for Mahler functions 1. Theorems 2. Proof of main theorem 3. Proof of multiplicity estimate Chapter 13. Algebraic Independence in Algebraic Groups. Part 1: Small Transcendence Degrees 1. Introduction 2. General statements 3. Concrete applications 4. A criteria 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 Chapter 14. Algebraic Independence in Algebraic Groups. Part 2: Large Transcendence Degrees 1. Introduction 2. Conjectures 3. Proofs Chapter 15. Some metric results in Transcendental Numbers Theory 1. Introduction 2. One dimensional results 3. Several dimensional results: 'comparison Theorem' 4. Several dimensional results: proof of Chudnovsky's conjecture Chapter 16. The Hilbert Nullstellensatz, Inequalities for Polynomials, and Algebraic Independence 1. The Hilbert Nullstellensatz and Effectivity 2. Liouville-Lojasiewicz Inequality 3. The Lojasiewicz Inequality Implies the Nullstellensatz 4. Geometric Version of the Nullstellensatz or Irrelevance of the Nullstellen Inequality for the Nullstellensatz 5. Arithmetic Aspects of the Bézout Version 6. Some Algorithmic Aspects of the Bézout Version Bibliography Index