Real Algebraic Geometry

1. Front Matter

   Pages I-IX

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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 17^th 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

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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. . . . . . . . . . . . . . . . . .

• 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

• 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