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E-grāmata: Real Algebraic Geometry

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Real Algebraic GeometryPalielināt
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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. . . . . . . . . . . . . . . . . .

Recenzijas

"Wie schon das franzosische Original, so ist auch die englische Version sehr sorgfaltig verfasst. Beim Vergleich zwischen beiden stosst man immer wieder auf Stellen, wo der ursprungliche Text nicht nur einfach ubersetzt, sondern im Interesse einer noch klareren Darstellung auch neu konzipiert worden ist. Kein Zweifel, dass die neue Fassung auch weiterhin die Rolle einer Standradreferenz spielen wird." DMV Jahresbericht, 103. Band, Heft 2, Juli 2001

Preface V
Introduction 1(6)
1. Ordered Fields, Real Closed Fields
7(16)
1.1 Ordered Fields, Real Fields
7(2)
1.2 Real Closed Fields
9(5)
1.3 Real Closure of an Ordered Field
14(3)
1.4 The Tarski-Seidenberg Principle
17(6)
2. Semi-algebraic Sets
23(36)
2.1 Algebraic and Semi-algebraic Sets
23(3)
2.2 Projection of Semi-algebraic Sets. Semi-algebraic Mappings
26(4)
2.3 Decomposition of Semi-algebraic Sets
30(4)
2.4 Connectedness
34(1)
2.5 Closed and Bounded Semi-algebraic Sets. Curve-selection Lemma
35(7)
2.6 Continuous Semi-algebraic Functions. Lojasiewicz's Inequality
42(4)
2.7 Separation of Closed Semi-algebraic Sets
46(4)
2.8 Dimension of Semi-algebraic Sets
50(4)
2.9 Some Analysis over a Real Closed Field
54(5)
3. Real Algebraic Varieties
59(24)
3.1 Real and Complex Algebraic Sets
59(3)
3.2 Real Algebraic Varieties
62(3)
3.3 Nonsingular Points
65(5)
3.4 Projective Spaces and Grassmannians
70(6)
3.5 Some Useful Constructions
76(7)
4. Real Algebra
83(14)
4.1 The Artin-Lang Homomorphism Theorem and the Real Nullstellensatz
83(3)
4.2 Cones, Convex Ideals
86(2)
4.3 Prime Cones
88(2)
4.4 The Positivstellensatz
90(4)
4.5 Real Principal Ideals
94(3)
5. The Tarski-Seidenberg Principle as a Transfer Tool
97(6)
5.1 Extension of Semi-algebraic Sets
97(1)
5.2 The Full Strength of the Tarski-Seidenberg Principle
98(2)
5.3 Further Results on Extension of Semi-algebraic Sets and Mappings
100(3)
6. Hilbert's 17(th) Problem. Quadratic Forms
103(30)
6.1 Solution of Hilbert's 17(th) Problem
103(3)
6.2 The Equivariant Version of Hilbert's 17(th) Problem
106(5)
6.3 Hilbert's Theorem about Positive Forms
111(3)
6.4 Quantitative Aspects of Hilbert's 17(th) Problem
114(8)
6.5 A Bound on the Number of Inequalities
122(6)
6.6 Bibliographic and Historical Notes
128(5)
7. Real Spectrum
133(28)
7.1 Definition and General Properties of the Real Spectrum
133(9)
7.2 Real Spectrum of a Ring of Polynomial Functions
142(4)
7.3 Semi-algebraic Functions on the Real Spectrum
146(3)
7.4 Semi-algebraic Families of Sets and Mappings
149(5)
7.5 Semi-algebraically Connected Components. Dimension
154(3)
7.6 Orderings and Central Points
157(4)
8. Nash Functions
161(46)
8.1 Germs of Nash Functions and Algebraic Power Series
161(6)
8.2 Local Properties of Nash Functions
167(4)
8.3 Approximation of Formal Solutions of a System of Nash Equations
171(1)
8.4 The Artin-Mazur Description of Nash Functions
172(3)
8.5 The Substitution Theorem. The Positivstellensatz for Nash Functions
175(3)
8.6 Nash Sets, Germs of Nash Sets
178(6)
8.7 Henselian Properties. Noetherian Property
184(8)
8.8 Efroymson's Approximation Theorem
192(5)
8.9 Tubular Neighbourhood. Extension Theorem
197(5)
8.10 Families of Nash Functions
202(5)
9. Stratifications
207(38)
9.1 Stratifying Families of Polynomials
207(9)
9.2 Triangulation of Semi-algebraic Sets
216(5)
9.3 Semi-algebraic Triviality of Semi-algebraic Mappings
221(6)
9.4 Triangulation of Semi-algebraic Functions
227(5)
9.5 Half-branches of Algebraic Curves
232(3)
9.6 The Theorems of Sard and Bertini
235(1)
9.7 Whitney's Conditions a and b
236(9)
10. Real Places
245(18)
10.1 Real Places and Orderings
245(4)
10.2 Real Places and Specialization in the Real Spectrum
249(5)
10.3 Half-branches of Algebraic Curves Again
254(2)
10.4 Fans and Basic Semi-algebraic Sets
256(7)
11. Topology of Real Algebraic Varieties
263(34)
11.1 Combinatorial Properties of Algebraic Sets
264(2)
11.2 Local Euler-Poincare Characteristic of Algebraic Sets
266(5)
11.3 Fundamental Class of a Real Algebraic Variety. Algebraic Homology
271(7)
11.4 Injective Regular Self-Mappings of an Algebraic Set
278(3)
11.5 Upper Bound for the Sum of the Betti Numbers of an Algebraic Set
281(4)
11.6 Nonsingular Algebraic Curves in the Real Projective Plane
285(5)
11.7 Appendix: Homology of Semi-algebraic Sets over a Real Closed Field
290(7)
12. Algebraic Vector Bundles
297(42)
12.1 Algebraic Vector Bundles
297(9)
12.2 Algebraic Line Bundles and the Divisor Class Group
306(2)
12.3 Approximation of Continuous Sections by Algebraic Sections
308(4)
12.4 Algebraic Approximation of C(XXX) Hypersurfaces
312(8)
12.5 Vector Bundles over Algebraic Curves and Surfaces
320(5)
12.6 Algebraic C-vector Bundles
325(6)
12.7 Nash Vector Bundles and Semi-algebraic Vector Bundles
331(8)
13. Polynomial or Regular Mappings with Values in Spheres
339(34)
13.1 Polynomial Mappings from S(n) into S(k)
339(7)
13.2 Hopf Forms and Nonsingular Bilinear Forms
346(6)
13.3 Approximation of Mappings with Values in S(1), S(2) or S(4)
352(9)
13.4 Homotopy Classes of Mappings into S(n)
361(7)
13.5 Mappings from a Product of Spheres into a Sphere
368(5)
14. Algebraic Models of C(XXX) Manifolds
373(10)
14.1 Algebraic Models of C(XXX) Manifolds
373(7)
14.2 More about the Topology of Real Algebraic Sets
380(3)
15. Witt Rings in Real Algebraic Geometry
383(24)
15.1 K(0) and the Witt Ring
383(9)
15.2 Separation of Connected Components by Signatures
392(7)
15.3 Comparison between W(P(V)) and W(S(0(V))
399(8)
Bibliography 407(14)
Index of Notation 421(6)
Index 427
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