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Real Algebraic Geometry

  • Jacek Bochnak
  • Michel Coste
  • Marie-Françoise Roy

Table of contents

  1. Front Matter
    Pages I-IX
  2. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 1-5
  3. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 7-21
  4. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 23-58
  5. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 59-82
  6. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 83-95
  7. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 97-102
  8. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 103-131
  9. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 133-160
  10. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 161-206
  11. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 207-243
  12. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 245-262
  13. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 263-296
  14. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 297-337
  15. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 339-371
  16. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 373-381
  17. Jacek Bochnak, Michel Coste, Marie-Françoise Roy
    Pages 383-405
  18. Back Matter
    Pages 407-430

About this book

Introduction

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

  • Jacek Bochnak
    • 1
  • Michel Coste
    • 2
  • Marie-Françoise Roy
    • 2
  1. 1.Mathematisch InstituutVrije UniversiteitAmsterdamThe Netherlands
  2. 2.Institut Mathématique de RennesUniversité de Rennes 1Rennes cedexFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-03718-8
  • Copyright Information Springer-Verlag Berlin Heidelberg 1998
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-08429-4
  • Online ISBN 978-3-662-03718-8
  • Series Print ISSN 0071-1136
  • Buy this book on publisher's site