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Real algebraic geometry:

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Bibliographic Details
Main Authors:	Bochnak, Jacek 1943- (Author), Coste, Michel 1950- (Author), Roy, Marie-Françoise 1950- (Author)
Format:	Book
Language:	English
Published:	Berlin Springer Berlin 2010
Edition:	1. ed., [softcover version of original hardcover edition 1998]
Series:	Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 36
Subjects:	Reelle algebraische Geometrie Algebraische Geometrie
Links:	http://deposit.dnb.de/cgi-bin/dokserv?id=3604059&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025104007&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Physical Description:	IX, 430 S. Ill., graph. Darst.
ISBN:	9783642084294 364208429X

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Record in the Search Index

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adam_text	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 .................. 46 2.8 Dimension of Semi-algebraic Sets ......................... 50 2.9 Some Analysis over a Real Closed Field ................... 54 3. Real Algebraic Varieties .................................. 59 3.1 Real and Complex Algebraic Sets ......................... 59 3.2 Real Algebraic Varieties ................................. 62 3.3 Nonsingular Points ..................................... 65 3.4 Projective Spaces and Grassmannians ..................... 70 3.5 Some Useful Constructions .............................. 76 4. Real Algebra ............................................. 83 4.1 The Artin-Lang Homomorphism Theorem and the Real Null- stellensatz ............................................. 83 4.2 Cones, Convex Ideals ................................... 86 4.3 Prime Cones ........................................... 88 4Л The Positivstellensatz................................... 90 4.5 Real Principal Ideals .................................... 94 5. The Tarski-Seidenberg Principle as a Transfer Tool ....... 97 5.1 Extension of Semi-algebraic Sets ......................... 97 5.2 The Full Strength of the Tarski-Seidenberg Principle ........ 98 5.3 Further Results on Extension of Semi-algebraic Sets and Map¬ pings .................................................100 6. Hubert s 17th Problem. Quadratic Forms .................103 6.1 Solution of Hubert s 17th Problem ........................103 6.2 The Equivariant Version of Hubert s 17th Problem ..........106 6.3 Hubert s Theorem about Positive Forms ................... Ill 6.4 Quantitative Aspects of Hubert s 17th Problem .............114 6.5 A Bound on the Number of Inequalities ...................122 6.6 Bibliographic and Historical Notes ........................128 7. Real Spectrum ...........................................133 7.1 Definition and General Properties of the Real Spectrum .....133 7.2 Real Spectrum of a Ring of Polynomial Functions ..........142 7.3 Semi-algebraic Functions on the Real Spectrum ............146 7.4 Semi-algebraic Families of Sets and Mappings ..............149 7.5 Semi-algebraically Connected Components. Dimension ......154 7.6 Orderings and Central Points ............................157 8. Nash Functions ...........................................161 8.1 Germs of Nash Functions and Algebraic Power Series ....... 161 8.2 Local Properties of Nash Functions ....................... 167 8.3 Approximation of Formal Solutions of a System of Nash Equations ............................................. 171 8.4 The Artin-Mazur Description of Nash Functions ............ 172 8.5 The Substitution Theorem. The Positivstellensatz for Nash Functions .............................................. 175 8.6 Nash Sets, Germs of Nash Sets ........................... 178 8.7 Henselian Properties. Noetherian Property ................. 184 8.8 Efroymson s Approximation Theorem ..................... 192 8.9 Tubular Neighbourhood. Extension Theorem ............... 197 8.10 Families of Nash Functions .............................. 202 9. Stratifications ............................................207 9.1 Stratifying Families of Polynomials .......................207 9.2 Triangulation of Semi-algebraic Sets ......................216 9.3 Semi-algebraic Triviality of Semi-algebraic Mappings ........221 9.4 Triangulation of Semi-algebraic Functions .................227 9.5 Half-branches of Algebraic Curves ........................232 9.6 The Theorems of Sard and Bertini ........................235 9.7 Whitney s Conditions о and 6............................236 10. Real Places ...............................................245 10.1 Real Places and Orderings ...............................245 10.2 Real Places and Specialization in the Real Spectrum ........249 10.3 Half-branches of Algebraic Curves Again ..................254 10.4 Fans and Basic Semi-algebraic Sets .......................256 11. Topology of Real Algebraic Varieties .....................263 11.1 Combinatorial Properties of Algebraic Sets ................264 11.2 Local Euler-Poincaré Characteristic of Algebraic Sets .......266 11.3 Fundamental Class of a Real Algebraic Variety. Algebraic Ho- mology ................................................271 11.4 Injective Regular Self-Mappings of an Algebraic Set .........278 11.5 Upper Bound for the Sum of the Betti Numbers of an Alge¬ braic Set ..............................................281 11.6 Nonsingular Algebraic Curves in the Real Projective Plane .. 285 11.7 Appendix: Homology of Semi-algebraic Sets over a Real Closed Field ..................................................290 12. Algebraic Vector Bundles .................................297 12.1 Algebraic Vector Bundles ................................297 12.2 Algebraic Line Bundles and the Divisor Class Group ........306 12.3 Approximation of Continuous Sections by Algebraic Sections . 308 12.4 Algebraic Approximation of C°° Hypersurfaces .............312 12.5 Vector Bundles over Algebraic Curves and Surfaces .........320 12.6 Algebraic С -vector Bundles ..............................325 12.7 Nash Vector Bundles and Semi-algebraic Vector Bundles .... 331 13. Polynomial or Regular Mappings with Values in Spheres . 339 13.1 Polynomial Mappings from Sn into Sk ....................339 13.2 Hopf Forms and Nonsingular Bilinear Forms ...............346 13.3 Approximation of Mappings with Values in S1, S2 or S4 .... 352 13.4 Homotopy Classes of Mappings into Sn ...................361 13.5 Mappings from a Product of Spheres into a Sphere .........368 14. Algebraic Models of C°° Manifolds ........................373 14.1 Algebraic Models of C°° Manifolds ........................373 14.2 More about the Topology of Real Algebraic Sets ............380 15. Witt Rings in Real Algebraic Geometry ..................383 15.1 Ko and the Witt Ring ..................................383 15.2 Separation of Connected Components by Signatures ........392 15.3 Comparison between W{V{V)) and W(S°{V)) .............399 Bibliography ..................................................407 Index of Notation ............................................421 Index .........................................................427
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series	Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge
series2	Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge
spellingShingle	Bochnak, Jacek 1943- Coste, Michel 1950- Roy, Marie-Françoise 1950- Real algebraic geometry Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge Reelle algebraische Geometrie (DE-588)4192004-1 gnd Algebraische Geometrie (DE-588)4001161-6 gnd
subject_GND	(DE-588)4192004-1 (DE-588)4001161-6
title	Real algebraic geometry
title_auth	Real algebraic geometry
title_exact_search	Real algebraic geometry
title_full	Real algebraic geometry Jacek Bochnak ; Michel Coste ; Marie-Francoise Roy
title_fullStr	Real algebraic geometry Jacek Bochnak ; Michel Coste ; Marie-Francoise Roy
title_full_unstemmed	Real algebraic geometry Jacek Bochnak ; Michel Coste ; Marie-Francoise Roy
title_short	Real algebraic geometry
title_sort	real algebraic geometry
topic	Reelle algebraische Geometrie (DE-588)4192004-1 gnd Algebraische Geometrie (DE-588)4001161-6 gnd
topic_facet	Reelle algebraische Geometrie Algebraische Geometrie
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volume_link	(DE-604)BV000899194
work_keys_str_mv	AT bochnakjacek realalgebraicgeometry AT costemichel realalgebraicgeometry AT roymariefrancoise realalgebraicgeometry

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