Algorithms in real algebraic geometry:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin
Springer Berlin
2010
|
| Edition: | 2., nd ed. Softcover version of original hardcover edition 2006 |
| Series: | Algorithms and Computation in Mathematics
10 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | X, 662 S. 40 schw.-w. Ill. 235 mm x 155 mm |
| ISBN: | 9783642069642 3642069649 |
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MARC
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| 100 | 1 | |a Basu, Saugata |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Algorithms in real algebraic geometry |c Saugata Basu ; Richard Pollack ; Marie-Françoise Coste-Roy |
| 250 | |a 2., nd ed. Softcover version of original hardcover edition 2006 | ||
| 264 | 1 | |a Berlin |b Springer Berlin |c 2010 | |
| 300 | |a X, 662 S. |b 40 schw.-w. Ill. |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Algorithms and Computation in Mathematics |v 10 | |
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| 689 | 0 | |8 1\p |5 DE-604 | |
| 700 | 1 | |a Pollack, Richard |e Verfasser |4 aut | |
| 700 | 1 | |a Coste-Roy, Marie-Françoise |e Verfasser |4 aut | |
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Record in the Search Index
| _version_ | 1805146753110900736 |
|---|---|
| adam_text |
Table
of
Contents
Introduction
. 1
1
Algebraically Closed Fields
. 11
1.1
Definitions and First Properties
. 11
1.2
Euclidean Division and Greatest Common Divisor
. 14
1.3
Projection Theorem for
Constructible
Sets
. 20
1.4
Quantifier Elimination and the Transfer Principle
. 25
1.5
Bibliographical Notes
. 27
2
Real Closed Fields
. 29
2.1
Ordered, Real and Real Closed Fields
. 29
2.2
Real Root Counting
. 44
2.2.1
Descartes's Law of Signs and the Budan-Fourier The¬
orem
. 44
2.2.2
Sturm's Theorem and the Cauchy Index
. 52
2.3
Projection Theorem for Algebraic Sets
. 57
2.4
Projection Theorem for Semi-Algebraic Sets
. 63
2.5
Applications
. 69
2.5.1
Quantifier Elimination and the Transfer Principle
. . 69
2.5.2
Semi-Algebraic Functions
. 71
2.5.3
Extension of Semi-Algebraic Sets and Functions
. 72
2.6
Puiseux Series
. 74
2.7
Bibliographical Notes
. 81
3
Semi-Algebraic Sets
. 83
3.1
Topology
. 83
3.2
Semi-algebraically Connected Sets
. 86
3.3
Semi-algebraic Germs
. 87
3.4
Closed and Bounded Semi-algebraic Sets
. 93
3.5
Implicit Function Theorem
. 94
3.6
Bibliographical Notes
. 99
Algebra
. 101
4.1
Discriminant and
Subdiscriminant
. 101
4.2
Resultant and Subresultant Coefficients
. 105
4.2.1
Resultant
. 105
4.2.2
Subresultant Coefficients
. 110
4.2.3
Subresultant Coefficients and Cauchy Index
. 113
4.3
Quadratic Forms and Root Counting
. 119
4.3.1
Quadratic Forms
. 119
4.3.2
Hermite's Quadratic Form
. 127
4.4
Polynomial Ideals
. 132
4.4.1
Hubert's Basis Theorem
. 132
4.4.2
Hubert's
Nullstellensatz. 136
4.5
Zero-dimensional Systems
. 143
4.6
Multivariate Hermite's Quadratic Form
. 149
4.7
Projective
Space and a Weak
Bézouťs
Theorem
. 153
4.8
Bibliographical Notes
. 157
Decomposition of Semi-Algebraic Sets
. 159
5.1
Cylindrical Decomposition
. 159
5.2
Semi-algebraically Connected Components
. 168
5.3
Dimension
. 170
5.4
Semi-algebraic Description of Cells
. 172
5.5
Stratification
. 174
5.6
Simplicial Complexes
. 181
5.7 Triangulation . 183
5.8
Hardt's Triviality Theorem and Consequences
. 186
5.9
Semi-algebraic Sard's Theorem
. 191
5.10
Bibliographical Notes
. 194
Elements of Topology
. 195
6.1
Simplicial Homology Theory
. 195
6.1.1
The Homology Groups of a Simplicial Complex
. 195
6.1.2
Simplicial Cohomology Theory
. 199
6.1.3
A Characterization of H1 in a Special Case
. 201
6.1.4
The Mayer-Vietoris Theorem
. 206
6.1.5
Chain
Homotopy
. 209
6.1.6
The Simplicial Homology Groups Are Invariant Under
Homeomorphism
. 213
6.2
Simplicial Homology of Closed and Bounded Semi-algebraic
Sets
. 221
6.2.1
Definitions and First Properties
. 221
6.2.2
Homotopy
. 223
6.3
Homology of Certain Locally Closed Semi-Algebraic Sets
. . 226
6.3.1
Homology of Closed Semi-algebraic Sets and of Sign Con¬
ditions
. 226
6.3.2
Homology of a Pair
. 228
6.3.3
Borel-Moore Homology
. 231
6.3.4
Euler-Poincaré
Characteristic
. 234
6.4
Bibliographical Notes
. 236
Quantitative Semi-algebraic Geometry
. 237
7.1
Morse Theory
. 237
7.2
Sum of the
Betti
Numbers of Real Algebraic Sets
. 256
7.3
Bounding the
Betti
Numbers of Realizations of Sign Conditions
. 262
7.4
Sum of the
Betti
Numbers of Closed Semi-algebraic Sets
. . 268
7.5
Sum of the
Betti
Numbers of Semi-algebraic Sets
. 273
7.6
Bibliographical Notes
. 280
Complexity of Basic Algorithms
. 281
8.1
Definition of Complexity
. 281
8.2
Linear Algebra
. 292
8.2.1
Size of Determinants
. 292
8.2.2
Evaluation of Determinants
. 294
8.2.3
Characteristic Polynomial
. 299
8.2.4
Signature of Quadratic Forms
. 300
8.3
Remainder Sequences and Subresultants
. 301
8.3.1
Remainder Sequences
. 301
8.3.2
Signed Subresultant Polynomials
. 303
8.3.3
Structure Theorem for Signed Subresultants
. 307
8.3.4
Size of Remainders and Subresultants
. 314
8.3.5
Specialization Properties of Subresultants
. 316
8.3.6
Subresultant Computation
. 317
8.4
Bibliographical Notes
. 322
9
Cauchy Index and Applications
. 323
9.1
Cauchy Index
. 323
9.1.1
Computing the Cauchy Index
. 323
9.1.2
Bezoutian and Cauchy Index
. 326
9.1.3
Signed Subresultant Sequence and Cauchy Index on an
Interval
. 330
9.2
Hankel Matrices
. 333
9.2.1
Hankel Matrices and Rational Functions
. 334
9.2.2
Signature of Hankel Quadratic Forms
. 337
9.3
Number of Complex Roots with Negative Real Part
. 344
9.4
Bibliographical Notes
. 350
10
Real Roots
. 351
10.1
Bounds on Roots
. 351
10.2
Isolating Real Roots
. 360
10.3
Sign Determination
. 383
10.4
Roots in a Real Closed Field
. 397
10.5
Bibliographical Notes
. 401
11
Cylindrical Decomposition Algorithm
. 403
11.1
Computing the Cylindrical Decomposition
. 404
11.1.1
Outline of the Method
. 404
11.1.2
Details of the Lifting Phase
. 408
11.2
Decision Problem
. 415
11.3
Quantifier Elimination
. 423
11.4
Lower Bound for Quantifier Elimination
. 426
11.5
Computation of Stratifying Families
. 428
11.6
Topology of Curves
. 430
11.7
Restricted Elimination
. 440
11.8
Bibliographical Notes
. 444
12
Polynomial System Solving
. 445
12.1
A Few Results on
Gröbner
Bases
. 445
12.2
Multiplication Tables
. 451
12.3
Special Multiplication Table
. 456
12.4
Univariate Representation
. 462
12.5
Limits of the Solutions of a Polynomial System
. 471
12.6
Finding Points in Connected Components of Algebraic Sets
. 483
12.7
Triangular Sign Determination
. 495
12.8 Computing
the
Euler-Poincaré
Characteristic of an Algebraic
Set
. 498
12.9
Bibliographical Notes
. 503
13
Existential Theory of the Reals
. 505
13.1
Finding Realizable Sign Conditions
. 506
13.2
A Few Applications
. 516
13.3
Sample Points on an Algebraic Set
. 519
13.4
Computing the
Euler-Poincaré
Characteristic of Sign Condi¬
tions
. 528
13.5
Bibliographical Notes
. 532
14
Quantifier Elimination
. 533
14.1
Algorithm for the General Decision Problem
. 534
14.2
Quantifier Elimination
. 547
14.3
Local Quantifier Elimination
. 551
14.4
Global Optimization
. 557
14.5
Dimension of Semi-algebraic Sets
. 558
14.6
Bibliographical Notes
. 562
15
Computing Roadmaps and Connected Components of Alge¬
braic Sets
. 563
15.1
Pseudo-critical Values and Connectedness
. 564
15.2
Roadmap of an Algebraic Set
. 568
15.3
Computing Connected Components of Algebraic Sets
. 580
15.4
Bibliographical Notes
. 592
16
Computing Roadmaps and Connected Components of Semi-
algebraic Sets
. 593
16.1
Special Values
. 593
16.2
Uniform Roadmaps
. 601
16.3
Computing Connected Components of Sign Conditions
. . . 608
16.4
Computing Connected Components of a Semi-algebraic Set
. 614
16.5
Roadmap Algorithm
. 617
16.6
Computing the First
Betti
Number of Semi-algebraic Sets
. 627
16.7
Bibliographical Notes
. 633
References
. 635
Index of Notation
. 645
Index
. 655 |
| any_adam_object | 1 |
| author | Basu, Saugata Pollack, Richard Coste-Roy, Marie-Françoise |
| author_facet | Basu, Saugata Pollack, Richard Coste-Roy, Marie-Françoise |
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| building | Verbundindex |
| bvnumber | BV040241453 |
| ctrlnum | (OCoLC)796273134 (DE-599)DNB1008496448 |
| discipline | Mathematik |
| edition | 2., nd ed. Softcover version of original hardcover edition 2006 |
| format | Book |
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| id | DE-604.BV040241453 |
| illustrated | Illustrated |
| indexdate | 2024-07-21T00:34:54Z |
| institution | BVB |
| isbn | 9783642069642 3642069649 |
| language | English |
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| physical | X, 662 S. 40 schw.-w. Ill. 235 mm x 155 mm |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Springer Berlin |
| record_format | marc |
| series | Algorithms and Computation in Mathematics |
| series2 | Algorithms and Computation in Mathematics |
| spelling | Basu, Saugata Verfasser aut Algorithms in real algebraic geometry Saugata Basu ; Richard Pollack ; Marie-Françoise Coste-Roy 2., nd ed. Softcover version of original hardcover edition 2006 Berlin Springer Berlin 2010 X, 662 S. 40 schw.-w. Ill. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Algorithms and Computation in Mathematics 10 Reelle algebraische Geometrie (DE-588)4192004-1 gnd rswk-swf Computeralgebra (DE-588)4010449-7 gnd rswk-swf Reelle algebraische Geometrie (DE-588)4192004-1 s Computeralgebra (DE-588)4010449-7 s 1\p DE-604 Pollack, Richard Verfasser aut Coste-Roy, Marie-Françoise Verfasser aut Algorithms and Computation in Mathematics 10 (DE-604)BV011131286 10 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3604927&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=025097650&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 |
| spellingShingle | Basu, Saugata Pollack, Richard Coste-Roy, Marie-Françoise Algorithms in real algebraic geometry Algorithms and Computation in Mathematics Reelle algebraische Geometrie (DE-588)4192004-1 gnd Computeralgebra (DE-588)4010449-7 gnd |
| subject_GND | (DE-588)4192004-1 (DE-588)4010449-7 |
| title | Algorithms in real algebraic geometry |
| title_auth | Algorithms in real algebraic geometry |
| title_exact_search | Algorithms in real algebraic geometry |
| title_full | Algorithms in real algebraic geometry Saugata Basu ; Richard Pollack ; Marie-Françoise Coste-Roy |
| title_fullStr | Algorithms in real algebraic geometry Saugata Basu ; Richard Pollack ; Marie-Françoise Coste-Roy |
| title_full_unstemmed | Algorithms in real algebraic geometry Saugata Basu ; Richard Pollack ; Marie-Françoise Coste-Roy |
| title_short | Algorithms in real algebraic geometry |
| title_sort | algorithms in real algebraic geometry |
| topic | Reelle algebraische Geometrie (DE-588)4192004-1 gnd Computeralgebra (DE-588)4010449-7 gnd |
| topic_facet | Reelle algebraische Geometrie Computeralgebra |
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