Real Algebraic Geometry: Proceedings of the Conference held in Rennes, France, June 24-28, 1991
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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. . . . . . . . . . . . . . . . . . Paperback, Label: Springer-Verlag, Springer-Verlag, Produktgruppe: Book, Publiziert: 1992-12, Studio: Springer-Verlag, Verkaufsrang: 13888126.

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Language: French. Brand new Book. Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contributions by: S. Akbulut and H. King; C. Andradas and J. Ruiz; A. Borobia; L. Br|cker; G.W. Brumfield; A. Castilla; Z. Charzynski and P. Skibinski; M. Coste and M. Reguiat; gtyarev; Z. Denkowska; J.-P. Francoise and F. Ronga; J.M. Gamboa and C. Ueno; D. Gondard- Cozette; I.V. Itenberg; P. Jaworski; A. Korchagin; T. Krasinksi and S. Spodzieja; K. Kurdyka; H. Lombardi; M. Marshall and L. Walter; V.F. Mazurovskii; G. Mikhalkin; T. Mostowski and E. Rannou; E.I. Shustin; N. Vorobjov.

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Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contributions by: S. Akbulut and H. King C. Andradas and J. Ruiz A. Borobia L. Br cker G.W. Brumfield A. Castilla Z. Charzynski and P. Skibinski M. Coste and M. Reguiat A. Degtyarev Z. Denkowska J.-P. Francoise and F. Ronga J.M. Gamboa and C. Ueno D. Gondard- Cozette I.V. Itenberg P. Jaworski A. Korchagin T. Krasinksi and S. Spodzieja K. Kurdyka H. Lombardi M. Marshall and L. Walter V.F. Mazurovskii G. Mikhalkin T. Mostowski and E. Rannou E.I. Shustin N. Vorobjov. 2008. viii, 420 S. VIII, 420 p. 235 mm Versandfertig in 6-10 Tagen, Softcover, Neuware, Offene Rechnung (Vorkasse vorbehalten).

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Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contributions by: S. Akbulut and H. King C. Andradas and J. Ruiz A. Borobia L. Br cker G.W. Brumfield A. Castilla Z. Charzynski and P. Skibinski M. Coste and M. Reguiat A. Degtyarev Z. Denkowska J.-P. Francoise and F. Ronga J.M. Gamboa and C. Ueno D. Gondard- Cozette I.V. Itenberg P. Jaworski A. Korchagin T. Krasinksi and S. Spodzieja K. Kurdyka H. Lombardi M. Marshall and L. Walter V.F. Mazurovskii G. Mikhalkin T. Mostowski and E. Rannou E.I. Shustin N. Vorobjov. 2008. viii, 420 S. VIII, 420 p. 235 mm Versandfertig in 3-5 Tagen, Softcover, Neuware, offene Rechnung (Vorkasse vorbehalten).

Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contributions by: S. Akbulut and H. King, C. Andradas and J. Ruiz, A. Borobia, L. Br cker, G.W. Brumfield, A. Castilla, Z. Charzynski and P. Skibinski, M. Coste and M. Reguiat, A. Degtyarev, Z. Denkowska, J.-P. Francoise and F. Ronga, J.M. Gamboa and C. Ueno, D. Gondard- Cozette, I.V. Itenberg, P. Jaworski, A. Korchagin, T. Krasinksi and S. Spodzieja, K. Kurdyka, H. Lombardi, M. Marshall and L. Walter, V.F. Mazurovskii, G. Mikhalkin, T. Mostowski and E. Rannou, E.I. Shustin, N. Vorobjov.

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Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade -see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry,techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry.CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.-Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry.Further contributions by: S. Akbulut and H. King; C.Andradas and J. Ruiz; A. Borobia; L. Br|cker; G.W.Brumfield; A. Castilla; Z. Charzynski and P. Skibinski; M.Coste and M. Reguiat; A. Degtyarev; Z. Denkowska; J.-P.Francoise and F. Ronga; J.M. Gamboa and C. Ueno; D. Gondard-Cozette; I.V. Itenberg; P. Jaworski; A. Korchagin; T.Krasinksi and S. Spodzieja; K. Kurdyka; H. Lombardi; M.Marshall and L. Walter; V.F. Mazurovskii; G. Mikhalkin;T. Mostowski and E. Rannou; E.I. Shustin; N. Vorobjov.

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Ten years after the first Rennes international meeting onreal algebraic geometry, the second one looked at thedevelopments in the subject during the intervening decade -see the 6 survey papers listed below. Further contributionsfrom the participants on recent research covered realalgebra and geometry, topology of real algebraic varietiesand 16thHilbert problem, classical algebraic geometry,techniques in real algebraic geometry, algorithms in realalgebraic geometry, semialgebraic geometry, real analyticgeometry.CONTENTS: Survey papers: M. Knebusch: Semialgebraic topologyin the last ten years.- R. Parimala: Algebraic andtopological invariants of real algebraic varieties.-Polotovskii, G.M.: On the classification of decomposingplane algebraic curves.- Scheiderer, C.: Real algebra andits applications to geometry in the last ten years: somemajor developments and results.- Shustin, E.L.: Topology ofreal plane algebraic curves.- Silhol, R.: Moduli problems inreal algebraic geometry.Further contributions by: S. Akbulut and H. King; C.Andradas and J. Ruiz; A. Borobia; L. Br|cker; G.W.Brumfield; A. Castilla; Z. Charzynski and P. Skibinski; M.Coste and M. Reguiat; A. Degtyarev; Z. Denkowska; J.-P.Francoise and F. Ronga; J.M. Gamboa and C. Ueno; D. Gondard-Cozette; I.V. Itenberg; P. Jaworski; A. Korchagin; T.Krasinksi and S. Spodzieja; K. Kurdyka; H. Lombardi; M.Marshall and L. Walter; V.F. Mazurovskii; G. Mikhalkin;T. Mostowski and E. Rannou; E.I. Shustin; N. Vorobjov.

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