Bochnaks RAG Contents

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Ergebnisse der Mathematikun ihrer Grenzgebiete3 Folge

Series of Modern Surveysin Mathematics

Editorial oardE Bombieri, Princeton S Peferman, StanfordM Gromov, Bures-sur-Yvette J Jost, Leipzig

Volume 6

J Kollar, Salt Lake City, Utah H.W. Lenstra, Jr., BerkeleyP.-L. Lions, Paris R Remmert Managing Editor), MunsterW Schmid, Cambridge, Mass. J.Tits, Paris


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Springer Verlag Berlin Heidelberg GmbH


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Jacek ochnakMichel CosteMarie Fran


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Jacek BochnakMathematisch InstituutVrije UniversiteitDe Boelelaan lOBINL-10B1 HV AmsterdamThe Netherlandse-mail: [email protected] CosteInstitut Mathematique de RennesUniversite de Rennes 1Campus de BeaulieuF-35042 Rennes cedexFrancee-mail: [email protected];:oise RoyInstitut Mathematique de RennesUniversite de Rennes 1Campus de BeaulieuF-35042 Rennes cedexFrancee-mail: [email protected] present edition is a substantially revised and expanded English version of the book Geometrie algebrique reelle , originally published in French as Ergebnisse der Mathematik un ihrerGrenzgebiete J. Folge Vol. 12

Library of Congress Cataloging-in-Publication Data applied forDie Deutsche Bibliothek - CIP-EinheitsaufnahmeBochnak, Jacek:Real algebraic geometrylJacek Bochnak; Michel Coste; Marie-Fran,oise Roy. - Berlin; Heidelberg;New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Singapore; Tokyo:Springer, 1998(Ergebnisse der Mathematik und ilirer Grenzgebiete; Folge 3, Vol. 36)

Mathematics Subject Classification 1991):Primary: 14PXX. Secondary: 03ClO, nExx, 12J15, 14Fo5, 19E99,19G12, 32Co5, 55R50, 57NBo, 5BAo7 5BA 5ISSN 0071-1136ISBN 978-3-642-08429-4 ISBN 978-3-662-03718-8 (eBook)DOI 10.1007/978-3-662-03718-8This work is subject to copyright. All rights are reserved, whethe r the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other ways, and storage in data banks.Duplication of this publication or parts thereof is permitted only under the provisions of theGerman Copyright Law of September 9,1965, in its current version, and pennission for use mustalways be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.

Springer-Verlag Berlin Heidelberg 1998Originally published by Springer-Verlag Berlin Heidelberg New York in 1998.Solleover reprint of the hardcover 1st edition 1998Typesetting: Camera-ready copy produced by the authors output fileusing a Springer TEX macro packageSPIN 114 2 91 44 3111 54321 - Printed on acid-free paper


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reface

he 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. Therehave also been new insights into material already in the French edition. Manyof these advances and insights have been incorporated in this English versionof the book, so that it may be viewed as being substantially different fromthe original.

We wish to thank Michael Buchner for his careful reading of the text andfor his linguistic corrections and stylistic improvements. he initial Jb TEiX filewas prepared by Thierry van Effelterre.he three authors participate in the European research network Real

Algebraic and Analytic Geometry . The first author was partially supportedby NATO Collaborative Research Grant 960011.April 1998 Jacek BochnakMichel Coste

Marie Pranroise Roy


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ableof ontents

P r e f a c eIntroduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Ordered Fields Real Closed Fields . . . . . . . . . . . . . . . . . 71.1 Ordered Fields, Real Fields . . . . . . . . . . . . . . . . . . . . . . . 71.2 Real Closed Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Real Closure of an Ordered Field. . . . . . . . . . . . . . . . . . . . . . . .. 141.4 The Tarski-Seidenberg Principle. . . . . . . . . . . . . . . . . . . . . . . . .. 172. Semi-algebraic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 232.1 Algebraic and Semi-algebraic Sets. . . . . . . . . . . . . . . . . . . . . . .. 23

2.2 Projection of Semi-algebraic Sets. Semi-algebraic Mappings.. 262.3 Decomposition of Semi-algebraic Sets. . . . . . . . . . . . . . . . . . . .. 32.4 Connec t ednes s 342.5 Closed and Bounded Semi-algebraic Sets. Curve-selectionLemma 352.6 Continuous Semi-algebraic Functions. Lojasiewicz's Inequality 422.7 Separation of Closed Semi-algebraic Sets. . . . . . . . . . . . . . . . . . 462.8 Dimension of Semi-algebraic Sets. . . . . . .. . . . . . . . . . . . . . . . . . 52.9 Some Analysis over a Real Closed Field.. . . . . . . . . . . . . . . . . . 543. Real Algebraic Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 593.1 Real and Complex Algebraic Sets. . . . . . . . . . . . . . . . . . . . . . . . . 593.2 Real Algebraic Varieties. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. 623.3 Nonsingular Points 653.4 Projective Spaces and Grassmannians . . . . . . . . . . . . . . . . . . . .. 73.5 Some Useful Constructions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 764. Real Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 834.1 The Artin-Lang Homomorphism Theorem and the Real Null-

stellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 Cones, Convex Ideals. . . . . . .. . . . . . . . .. . . . . . . . . . . . . .. . . .. 864.3 Prime Cones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 884.4 The Positivstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94.5 Real Principal Ideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


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VIII Table of Contents

5. The Tarski-Seidenberg Principle as a Transfer Tool . . . . . . . 975.1 Extension of Semi-algebraic Sets 975.2 The Full Strength of the Tarski-Seidenberg Principle 985.3 Further Results on Extension of Semi-algebraic Sets and Map-pings 100

6. Hilbert s 17th Problem. Quadratic Forms 1036 1 Solution of Hilbert's 17th Problem 1036.2 The Equivariant Version of Hilbert's 1 h Problem 1066.3 Hilbert's Theorem about Positive Forms 1116.4 Quantitative Aspects of Hilbert's 1 h Problem 1146.5 A Bound on the Number of Inequalities 1226.6 Bibliographic and Historical Notes. . . . . . . . . . . . . . . . . . . . . . . . 128

7. Real Spectrum 1337 1 Definition and General Properties of the Real Spectrum 1337.2 Real Spectrum of a Ring of Polynomial Functions 1427.3 Semi-algebraic Functions on the Real Spectrum 1467.4 Semi-algebraic Families of Sets and Mappings 1497 5 Semi-algebraically Connected Components. Dimension 1547.6 Orderings and Central Points 157

8. Nash Functions 1618 1 Germs of Nash Functions and Algebraic Power Series 1618.2 Local Properties of Nash Functions 1678.3 Approximation of Formal Solutions of a System of NashEquations 1718.4 The Artin-Mazur Description of Nash Functions 1728.5 The Substitution Theorem. The Positivstellensatz for NashFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1758.6 Nash Sets, Germs of Nash Sets 1788.7 Henselian Properties. Noetherian Property. . . . . . . . . . . . . . . . . 1848.8 Efroymson's Approximation Theorem 1928.9 Tubular Neighbourhood. Extension Theorem 1978.10 Families of Nash Functions 202

9. Stratifications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2079 1 Stratifying Families of Polynomials 2079.2 Triangulation of Semi-algebraic Sets 2169.3 Semi-algebraic Triviality of Semi-algebraic Mappings 2219.4 Triangulation of Semi-algebraic Functions 2279.5 Half-branches of Algebraic Curves 2329.6 The Theorems of Sard and Bertini 2359.7 Whitney's Conditions and . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236


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Table of Contents IX

10 Real Places 24510.1 Real Places and Orderings 24510.2 Real Places and Specialization in the Real Spectrum 24910.3 Half-branches of Algebraic Curves Again 25410.4 Fans and Basic Semi-algebraic Sets 256

11 Topology of Real Algebraic Varieties , 26311.1 Combinatorial Properties of Algebraic Sets 26411.2 Local Euler-Poincare Characteristic of Algebraic Sets 26611.3 Fundamental Class of a Real Algebraic Variety. Algebraic Homology 27111.4 Injective Regular Self-Mappings of an Algebraic Set 27811.5 Upper Bound for the Sum of the Betti Numbers of an Alge-braic Set 28111.6 Nonsingular Algebraic Curves in the Real Projective Plane 28511.7 Appendix: Homology of Semi-algebraic Sets over a Real ClosedField , , 290

12 Algebraic Vector Bundles , 29712.1 Algebraic Vector Bundles 29712.2 Algebraic Line Bundles and the Divisor Class Group 30612.3 Approximation o ~ o n t i n u o u s Sections by Algebraic Sections. 30812.4 Algebraic Approximation of Coo Hypersurfaces 31212.5 Vector Bundles over Algebraic Curves and Surfaces 32012.6 Algebraic (>vector Bundles 32512.7 Nash Vector Bundles and Semi-algebraic Vector Bundles 331

13 Polynomial or Regular Mappings with Values in Spheres . 33913.1 Polynomial Mappings from 8 into k . . . . . . . . . . . . 33913.2 Hopf Forms and Nonsingular Bilinear Forms 34613.3 Approximation of Mappings with Values in 8 1, 8 2 or 8 4 35213.4 Homotopy Classes of Mappings into 8 36113.5 Mappings from a Product of Spheres into a Sphere 368

14 Algebraic Models of oo Manifolds 37314.1 Algebraic Models of Coo Manifolds 37314.2 More about the Topology of Real Algebraic Sets 38015 Witt Rings in Real Algebraic Geometry 38315.1 K o and the Witt Ring 38315.2 Separation of Connected Components by Signatures 39215.3 Comparison between W P V and W SO V 399Bibliography 407Index of Notation , 421Index 427

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