Introduction to algebraic independence theory

摘要: Preface List of Contributors Chapter 1. PHI (tau,z) and Transcendence Differential rings modular forms 2. Explicit differential equations 3. Singular values 4. on phi z Mahler's conjecture other transcendence results Introduction A proof K. Barre's work functions Conjectures about exponential Algebraic independence for Ramanujan Main theorem consequences How it can be proved? Constructions the sequence polynomials fundamentals 5. Another Theorem 1.1 Some remarks proofs algebraic Connection with elliptic series phi, ephi TAU ( 1/4) Approximation properties Elimination multihomogene Formes eliminantes des ideaux multihomogenes resultantes 6. Diophantine geometry theory Degree Height Geometric arithmetic Bezout theorems Distance from a point to variety Auxiliary 7. First metric 8. Second Geometrie diophantienne multiprojective Hauteurs Une formule d'intersection Distances Criteria Mixed Segre- Veronese embeddings Multi-projective criteria 9. Upper bounds (geometric ) Hilbert The absolute case (following Kollar) relative 10. Multiplicity estimates solutions Reduction tobounds polynomial ideals assertions End 2.2 D-property 11. Zero Estimates Commutative Groups an intersection group Translation derivations Statement zero estimate 12. Measures Mahler Theorems Proof main multiplicity 13. Independence in Groups. Part 1: Small Degrees General statements Concrete applications multiplicities Introducing matrix M rank Analytic upper bound Proposition 5.1 14. 2: Large Proofs 15. Transcendental Numbers Theory One dimensional Several results: 'comparison Theorem' Chudnovsky's 16. Nullstellensatz, Inequalities Polynomials, Nullstellensatz Effectivity Liouville-Lojasiewicz Inequality Lojasiewicz Implies Version or Irrelevance Nullstellen Arithmetic Aspects Algorithmic Bibliography Index