Real Algebraic Geometry
Автор(ы):  Bochnak J., Coste M.
06.10.2007

Год изд.:  1998 
Описание:  The present volume is a translation, revision and updating of the book (published in French) with the title "Geometrie Algebrique Reelle". Since its publication 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. 
Оглавление: 
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 TarskiSeidenberg Principle [17] 2. Semialgebraic Sets [23] 2.1 Algebraic and Semialgebraic Sets [23] 2.2 Projection of Semialgebraic Sets. Semialgebraic Mappings [26] 2.3 Decomposition of Semialgebraic Sets [30] 2.4 Connectedness [34] 2.5 Closed and Bounded Semialgebraic Sets. Curveselection Lemma [35] 2.6 Continuous Semialgebraic Functions. Lojasiewicz's Inequality [42] 2.7 Separation of Closed Semialgebraic Sets [46] 2.8 Dimension of Semialgebraic 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 ArtinLang Homomorphism Theorem and the Real Nullstellensatz [83] 4.2 Cones, Convex Ideals [86] 4.3 Prime Cones [88] 4.4 The Positivstellensatz [90] 4.5 Real Principal Ideals [94] 5. The TarskiSeidenberg Principle as a Transfer Tool [97] 5.1 Extension of Semialgebraic Sets [97] 5.2 The Full Strength of the TarskiSeidenberg Principle [98] 5.3 Further Results on Extension of Semialgebraic Sets and Mappings [100] 6. Hilbert's 17th Problem. Quadratic Forms [103] 6.1 Solution of Hilbert's 17th Problem [103] 6.2 The Equivariant Version of Hilbert's 17th Problem [106] 6.3 Hilbert's Theorem about Positive Forms [111] 6.4 Quantitative Aspects of Hilbert'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 Semialgebraic Functions on the Real Spectrum [146] 7.4 Semialgebraic Families of Sets and Mappings [149] 7.5 Semialgebraically 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 ArtinMazur 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 Semialgebraic Sets [216] 9.3 Semialgebraic Triviality of Semialgebraic Mappings [221] 9.4 Triangulation of Semialgebraic Functions [227] 9.5 Halfbranches of Algebraic Curves [232] 9.6 The Theorems of Sard and Bertini [235] 9.7 Whitney's Conditions a and b [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 Halfbranches of Algebraic Curves Again [254] 10.4 Fans and Basic Semialgebraic Sets [256] 11. Topology of Real Algebraic Varieties [263] 11.1 Combinatorial Properties of Algebraic Sets [264] 11.2 Local EulerPoincare Characteristic of Algebraic Sets [266] 11.3 Fundamental Class of a Real Algebraic Variety. Algebraic Homology [271] 11.4 Injective Regular SelfMappings of an Algebraic Set [278] 11.5 Upper Bound for the Sum of the Betti Numbers of an Algebraic Set [281] 11.6 Nonsingular Algebraic Curves in the Real Projective Plane [285] 11.7 Appendix: Homology of Semialgebraic 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 Cvector Bundles [325] 12.7 Nash Vector Bundles and Semialgebraic Vector Bundles [331] 13. Polynomial or Regular Mappings with Values in Spheres [339] 13.1 Polynomial Mappings from (?) into (?) [339] 13.2 Hopf Forms and Nonsingular Bilinear Forms [346] 13.3 Approximation of Mappings with Values in (?), (?) or (?) [352] 13.4 Homotopy Classes of Mappings into (?) [361] 13.5 Mappings from a Product of Spheres into a Sphere [368] 14. Algebraic Models of (?) Manifolds [373] 14.1 Algebraic Models of (?) Manifolds [373] 14.2 More about the Topology of Real Algebraic Sets [380] 15. Witt Rings in Real Algebraic Geometry [383] 15.1 (?) and the Witt Ring [333] 15.2 Separation of Connected Components by Signatures [392] 15.3 Comparison between W(?(?)) and W(?(?)) [399] Bibliography [407] Index of Notation [421] Index [427] 
Формат:  djvu 
Размер:  6039123 байт 
Язык:  ENG 
Рейтинг:  184 
 
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