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  • © 1998

Real Algebraic Geometry

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  • A systematic treatment of real algebraic geometry. -

  • Carefully and clearly written - May serve as a basis for a graduate course - Translation, revision and updating of ERGEBNISSE vol.

  • 12 published 1987 in French - Many of the new developments in the theory are included in this book

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  • ISBN: 978-3-662-03718-8
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Table of contents (16 chapters)

  1. Front Matter

    Pages I-IX
  2. Introduction

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 1-5
  3. Ordered Fields, Real Closed Fields

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 7-21
  4. Semi-algebraic Sets

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 23-58
  5. Real Algebraic Varieties

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 59-82
  6. Real Algebra

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 83-95
  7. The Tarski-Seidenberg Principle as a Transfer Tool

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 97-102
  8. Hilbert’s 17th Problem. Quadratic Forms in Real Algebraic Geometry

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 103-131
  9. Real Spectrum

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 133-160
  10. Nash Functions

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 161-206
  11. Stratifications

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 207-243
  12. Real Places

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 245-262
  13. Topology of Real Algebraic Varieties

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 263-296
  14. Algebraic Vector Bundles

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 297-337
  15. Polynomial or Regular Mappings with Values in Spheres

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 339-371
  16. Algebraic Models of C Manifolds

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 373-381
  17. Witt Rings in Real Algebraic Geometry

    • Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 383-405
  18. Back Matter

    Pages 407-430

About this book

The present volume is a translation, revision and updating of our book (pub­ lished in French) with the title "Geometrie Algebrique Reelle". Since its pub­ lication in 1987 the theory has made advances in several directions. There have also been new insights into material already in the French edition. Many of these advances and insights have been incorporated in this English version of the book, so that it may be viewed as being substantially different from the original. We wish to thank Michael Buchner for his careful reading of the text and for his linguistic corrections and stylistic improvements. The initial Jb. TEiX file was prepared by Thierry van Effelterre. The three authors participate in the European research network "Real Algebraic and Analytic Geometry". The first author was partially supported by NATO Collaborative Research Grant 960011. Jacek Bochnak April 1998 Michel Coste Marie-Pranroise Roy Table of Contents Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Ordered Fields, Real Closed Fields . . . . . . . . . . . . . . . . . . . . . . . 7 1. 1 Ordered Fields, Real Fields . . . . . " . . . . . . . . . . . . . . . . . . . . . . . 7 1. 2 Real Closed Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1. 3 Real Closure of an Ordered Field. . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 4 The Tarski-Seidenberg Principle. . . . . . . . . . . . . . . . . . . . . . . . . . 17 2. Semi-algebraic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2. 1 Algebraic and Semi-algebraic Sets. . . . . . . . . . . . . . . . . . . . . . . . 23 2. 2 Projection of Semi-algebraic Sets. Semi-algebraic Mappings. . 26 2. 3 Decomposition of Semi-algebraic Sets. . . . . . . . . . . . . . . . . . . . . 30 2. 4 Connectedness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2. 5 Closed and Bounded Semi-algebraic Sets. Curve-selection Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2. 6 Continuous Semi-algebraic Functions. Lojasiewicz's Inequality 42 2. 7 Separation of Closed Semi-algebraic Sets. . . . . . . . . . . . . . . . . .

Keywords

  • Nash functions
  • Nash-Funktionen
  • Semi-algebraic sets
  • Semi-algebraische Mengen
  • algebraic varieties
  • algebraic vector bundles
  • algebraische Vektorbündel
  • quadratische Formen
  • real algebraic varieties
  • reelle algebraische Mannigfaltigkeiten
  • singularity theory

Authors and Affiliations

  • Mathematisch Instituut, Vrije Universiteit, Amsterdam, The Netherlands

    Jacek Bochnak

  • Institut Mathématique de Rennes, Université de Rennes 1, Rennes cedex, France

    Michel Coste, Marie-Françoise Roy

Bibliographic Information

  • Book Title: Real Algebraic Geometry

  • Authors: Jacek Bochnak, Michel Coste, Marie-Françoise Roy

  • Series Title: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics

  • DOI: https://doi.org/10.1007/978-3-662-03718-8

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1998

  • Hardcover ISBN: 978-3-540-64663-1

  • Softcover ISBN: 978-3-642-08429-4

  • eBook ISBN: 978-3-662-03718-8

  • Series ISSN: 0071-1136

  • Series E-ISSN: 2197-5655

  • Edition Number: 1

  • Number of Pages: X, 430

  • Additional Information: Title of the original French edition: Geometrie algebrique reelle: in the series Ergebnisse Mathematik 3. Folge volume 12

  • Topics: Algebraic Geometry

Buying options

eBook
USD 169.00
Price excludes VAT (USA)
  • ISBN: 978-3-662-03718-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 219.99
Price excludes VAT (USA)
Hardcover Book
USD 219.99
Price excludes VAT (USA)