Real algebraic geometry:
Saved in:
| Main Authors: | , , |
|---|---|
| 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: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | IX, 430 S. Ill., graph. Darst. |
| ISBN: | 9783642084294 364208429X |
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MARC
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| 245 | 1 | 0 | |a Real algebraic geometry |c Jacek Bochnak ; Michel Coste ; Marie-Francoise Roy |
| 250 | |a 1. ed., [softcover version of original hardcover edition 1998] | ||
| 264 | 1 | |a Berlin |b Springer Berlin |c 2010 | |
| 300 | |a IX, 430 S. |b Ill., graph. Darst. | ||
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| 490 | 1 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge |v 36 | |
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Record in the Search Index
| _version_ | 1805146776454299648 |
|---|---|
| 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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| author_GND | (DE-588)120268256 (DE-588)12026823X (DE-588)120268221 |
| author_facet | Bochnak, Jacek 1943- Coste, Michel 1950- Roy, Marie-Françoise 1950- |
| author_role | aut aut aut |
| author_sort | Bochnak, Jacek 1943- |
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| discipline | Mathematik |
| edition | 1. ed., [softcover version of original hardcover edition 1998] |
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| institution | BVB |
| isbn | 9783642084294 364208429X |
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| physical | IX, 430 S. Ill., graph. Darst. |
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| series | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge |
| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge |
| spelling | Bochnak, Jacek 1943- Verfasser (DE-588)120268256 aut Real algebraic geometry Jacek Bochnak ; Michel Coste ; Marie-Francoise Roy 1. ed., [softcover version of original hardcover edition 1998] Berlin Springer Berlin 2010 IX, 430 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 36 Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Reelle algebraische Geometrie (DE-588)4192004-1 gnd rswk-swf Reelle algebraische Geometrie (DE-588)4192004-1 s 1\p DE-604 Algebraische Geometrie (DE-588)4001161-6 s 2\p DE-604 Coste, Michel 1950- Verfasser (DE-588)12026823X aut Roy, Marie-Françoise 1950- Verfasser (DE-588)120268221 aut Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 36 (DE-604)BV000899194 36 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3604059&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Passau application/pdf 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 Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Bochnak, Jacek 1943- Coste, Michel 1950- Roy, Marie-Françoise 1950- Real algebraic geometry Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge Algebraische Geometrie (DE-588)4001161-6 gnd Reelle algebraische Geometrie (DE-588)4192004-1 gnd |
| subject_GND | (DE-588)4001161-6 (DE-588)4192004-1 |
| 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 | Algebraische Geometrie (DE-588)4001161-6 gnd Reelle algebraische Geometrie (DE-588)4192004-1 gnd |
| topic_facet | Algebraische Geometrie Reelle algebraische Geometrie |
| url | 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 |
| volume_link | (DE-604)BV000899194 |
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