Introduction to Algebraic Independence Theory
(Sprache: Englisch)
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...
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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.
Klappentext zu „Introduction to Algebraic Independence Theory “
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.
Inhaltsverzeichnis zu „Introduction to Algebraic Independence Theory “
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. Barré'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. Élimination multihomogène 1. Introduction2. Formes éliminantes des idéaux multihomogènes 3. Formes résultantes des idéaux multihomogènesChapter 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 theoremChapter 7. Géométrie 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 Kollár) 2. The relative caseChapter 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 2.25. D-property for Ramanujan functionsChapter 11. Zero Estimates on Commutative Algebraic Groups 1. Introduction2. Degree of an
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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 Bézout Version6. Some Algorithmic Aspects of the Bézout VersionBibliographyIndex
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Bibliographische Angaben
- Autoren: M. Laurent , D. Roy , F. Amoroso , D. Bertrand , W. D. Brownawell , G. Diaz , Yu. V. Nesterenko , K. Nishioka , P. Philippon , G. Remond , M. Waldschmidt
- 2001, 276 Seiten, Maße: 15,5 x 23,3 cm, Kartoniert (TB), Englisch
- Herausgegeben: Patrice Philippon, Yuri V. Nesterenko
- Verlag: Springer Berlin Heidelberg
- ISBN-10: 3540414967
- ISBN-13: 9783540414964
- Erscheinungsdatum: 18.01.2001
Sprache:
Englisch
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