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:
Real Algebraic Geometry — обложка книги.
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]
    4.4 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 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 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 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 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-Poincare Characteristic of Algebraic Sets [266]
    11.3 Fundamental Class of a Real Algebraic Variety. Algebraic Homology [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 Algebraic 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 C-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 (?) 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
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