Algorithms in real algebraic geometry
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| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin
Springer Berlin
2010
|
| Ausgabe: | 2., nd ed. Softcover version of original hardcover edition 2006 |
| Schriftenreihe: | Algorithms and Computation in Mathematics
10 |
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| 015 | |a 10,N48 |2 dnb | ||
| 016 | 7 | |a 1008496448 |2 DE-101 | |
| 020 | |a 9783642069642 |c Pb. : EUR 69.50 (DE) (freier Pr.), sfr 101.00 (freier Pr.) |9 978-3-642-06964-2 | ||
| 020 | |a 3642069649 |c Pb. : EUR 69.50 (DE) (freier Pr.), sfr 101.00 (freier Pr.) |9 3-642-06964-9 | ||
| 024 | 3 | |a 9783642069642 | |
| 028 | 5 | 2 | |a Best.-Nr.: 12880997 |
| 035 | |a (OCoLC)796273134 | ||
| 035 | |a (DE-599)DNB1008496448 | ||
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| 041 | 0 | |a eng | |
| 044 | |a gw |c XA-DE-BE | ||
| 049 | |a DE-739 | ||
| 084 | |a 510 |2 sdnb | ||
| 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 | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Algorithms and Computation in Mathematics |v 10 | |
| 650 | 0 | 7 | |a Computeralgebra |0 (DE-588)4010449-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Reelle algebraische Geometrie |0 (DE-588)4192004-1 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Reelle algebraische Geometrie |0 (DE-588)4192004-1 |D s |
| 689 | 0 | 1 | |a Computeralgebra |0 (DE-588)4010449-7 |D s |
| 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 | |
| 830 | 0 | |a Algorithms and Computation in Mathematics |v 10 |w (DE-604)BV011131286 |9 10 | |
| 856 | 4 | 2 | |m X:MVB |q text/html |u http://deposit.dnb.de/cgi-bin/dokserv?id=3604927&prov=M&dok_var=1&dok_ext=htm |3 Inhaltstext |
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| 883 | 1 | |8 1\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-025097650 | |
Datensatz im Suchindex
| _version_ | 1819672724909326336 |
|---|---|
| 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 |
| author_role | aut aut aut |
| author_sort | Basu, Saugata |
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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 | 2024-12-24T02:43:17Z |
| institution | BVB |
| isbn | 9783642069642 3642069649 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025097650 |
| oclc_num | 796273134 |
| open_access_boolean | |
| owner | DE-739 |
| owner_facet | DE-739 |
| 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 |
| spellingShingle | Basu, Saugata Pollack, Richard Coste-Roy, Marie-Françoise Algorithms in real algebraic geometry Algorithms and Computation in Mathematics Computeralgebra (DE-588)4010449-7 gnd Reelle algebraische Geometrie (DE-588)4192004-1 gnd |
| subject_GND | (DE-588)4010449-7 (DE-588)4192004-1 |
| 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 | Computeralgebra (DE-588)4010449-7 gnd Reelle algebraische Geometrie (DE-588)4192004-1 gnd |
| topic_facet | Computeralgebra Reelle algebraische Geometrie |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3604927&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=025097650&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011131286 |
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