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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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    You can also search for this editor in PubMed   Google Scholar

  2. Patrice Philippon

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

    View editor publications

    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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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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Keywords

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.

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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