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Introduction to Algebraic Independence Theory

  • Book
  • © 2001

Overview

Editors:
  1. Yuri V. Nesterenko

    1. Faculty of Mechanics and Mathematics, Moscow University, Moscow, Russia

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  2. Patrice Philippon

    1. Institut de Mathématiques de Jussieu, UMR 7586 du CNRS, Paris Cedex 05, France

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    You can also search for this editor in PubMed   Google Scholar

  • The first didactically-conceived presentation of the recent development in transcendence theory
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 1752)

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About this book

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.

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Table of contents (16 chapters)

  1. Front Matter

    Pages I-XIII
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  2. Θ(τ, z) and Transcendence

    Pages 1-11

  3. Mahler’s conjecture and other transcendence Results

    Pages 13-26

  4. Algebraic independence for values of Ramanujan Functions

    Pages 27-46

  5. Some remarks on proofs of algebraic independence

    Pages 47-51

  6. Elimination multihomogene

    Pages 53-81

  7. Diophantine geometry

    Pages 83-94

  8. Géométrie diophantienne multiprojective

    Pages 95-131

  9. Criteria for algebraic independence

    Pages 133-141

  10. Upper bounds for (geometric) Hilbert functions

    Pages 143-148

  11. Multiplicity estimates for solutions of algebraic differential equations

    Pages 149-165

  12. Zero Estimates on Commutative Algebraic Groups

    Pages 167-185

  13. Measures of algebraic independence for Mahler functions

    Pages 187-197

  14. Algebraic Independence in Algebraic Groups. Part 1: Small Transcendence Degrees

    Pages 199-211

  15. Algebraic Independence in Algebraic Groups. Part II: Large Transcendence Degrees

    Pages 213-225

  16. Some metric results in Transcendental Numbers Theory

    Pages 227-237

  17. The Hilbert Nullstellensatz, Inequalities for Polynomials, and Algebraic Independence

    Pages 239-248

  18. Back Matter

    Pages 249-256
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Editors and Affiliations

  • Faculty of Mechanics and Mathematics, Moscow University, Moscow, Russia

    Yuri V. Nesterenko

  • Institut de Mathématiques de Jussieu, UMR 7586 du CNRS, Paris Cedex 05, France

    Patrice Philippon

Bibliographic Information

  • Book Title: Introduction to Algebraic Independence Theory

  • Editors: Yuri V. Nesterenko, Patrice Philippon

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/b76882

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2001

  • Softcover ISBN: 978-3-540-41496-4Published: 18 January 2001

  • eBook ISBN: 978-3-540-44550-0Published: 01 July 2003

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XVI, 260

  • Topics: Number Theory, Algebraic Geometry

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