Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
New York [u.a.]
Springer
1997
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Undergraduate texts in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 523 - 526 |
| Beschreibung: | XIII, 536 S. graph. Darst. |
| ISBN: | 0387946802 |
Internformat
MARC
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| 100 | 1 | |a Cox, David A. |e Verfasser |4 aut | |
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| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1997 | |
| 300 | |a XIII, 536 S. |b graph. Darst. | ||
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Datensatz im Suchindex
| _version_ | 1804125694170497024 |
|---|---|
| adam_text | Contents
Preface to the First Edition vii
Preface to the Second Edition ix
1. Geometry, Algebra, and Algorithms 1
§1. Polynomials and Affine Space 1
§2. Affine Varieties 5
§3. Parametrizations of Affine Varieties 14
§4. Ideals 29
§5. Polynomials of One Variable 37
2. Groebner Bases 47
§1. Introduction 47
§2. Orderings on the Monomials in k[x ,..., xn] 52
§3. A Division Algorithm in it [*i,..., xn] 59
§4. Monomial Ideals and Dickson s Lemma 67
§5. The Hilbert Basis Theorem and Groebner Bases 73
§6. Properties of Groebner Bases 79
§7. Buchberger s Algorithm 86
§8. First Applications of Groebner Bases 93
§9. (Optional) Improvements on Buchberger s Algorithm 99
3. Elimination Theory 112
§1. The Elimination and Extension Theorems 112
§2. The Geometry of Elimination 120
§3. Implicitization 124
§4. Singular Points and Envelopes 133
§5. Unique Factorization and Resultants 146
§6. Resultants and the Extension Theorem 158
xi
xii Contents
4. The Algebra Geometry Dictionary 167
§1. Hilbert s Nullstellensatz 167
§2. Radical Ideals and the Ideal Variety Correspondence 173
§3. Sums, Products, and Intersections of Ideals 180
§4. Zariski Closure and Quotients of Ideals 190
§5. Irreducible Varieties and Prime Ideals 195
§6. Decomposition of a Variety into Irreducibles 200
§7. (Optional) Primary Decomposition of Ideals 206
§8. Summary 210
5. Polynomial and Rational Functions on a Variety 212
§1. Polynomial Mappings 212
§2. Quotients of Polynomial Rings 218
§3. Algorithmic Computations in k[xi,..., xn]/I 226
§4. The Coordinate Ring of an Affine Variety 235
§5. Rational Functions on a Variety 245
§6. (Optional) Proof of the Closure Theorem 254
6. Robotics and Automatic Geometric Theorem Proving 261
§1. Geometric Description of Robots 261
§2. The Forward Kinematic Problem 267
§3. The Inverse Kinematic Problem and Motion Planning 274
§4. Automatic Geometric Theorem Proving 286
§5. Wu s Method 302
7. Invariant Theory of Finite Groups 311
§1. Symmetric Polynomials 311
§2. Finite Matrix Groups and Rings of Invariants 321
§3. Generators for the Ring of Invariants 329
§4. Relations Among Generators and the Geometry of Orbits 338
8. Projective Algebraic Geometry 349
§1. The Projective Plane 349
§2. Projective Space and Projective Varieties 360
§3. The Projective Algebra Geometry Dictionary 370
§4. The Projective Closure of an Affine Variety 378
§5. Projective Elimination Theory 384
§6. The Geometry of Quadric Hypersurfaces 399
§7. Bezout s Theorem 412
9. The Dimension of a Variety 429
§1. The Variety of a Monomial Ideal 429
§2. The Complement of a Monomial Ideal 433
Contents xiii
§3. The Hilbert Function and the Dimension of a Variety 446
§4. Elementary Properties of Dimension 457
§5. Dimension and Algebraic Independence 465
§6. Dimension and Nonsingularity 473
§7. The Tangent Cone 483
Appendix A. Some Concepts from Algebra 497
§1. Fields and Rings 497
§2. Groups 498
§3. Determinants 499
Appendix B. Pseudocode 501
§1. Inputs, Outputs, Variables, and Constants 501
§2. Assignment Statements 502
§3. Looping Structures 502
§4. Branching Structures 503
Appendix C. Computer Algebra Systems 505
§1. AXIOM 505
§2. Maple 508
§3. Mathematica 510
§4. REDUCE 512
§5. Other Systems 516
Appendix D. Independent Projects 518
§1. General Comments 518
§2. Suggested Projects 518
References 523
Index 527
|
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| author | Cox, David A. Little, John B. O'Shea, Donal 1952- |
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| author_role | aut aut aut |
| author_sort | Cox, David A. |
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| bvnumber | BV011196869 |
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| callnumber-subject | QA - Mathematics |
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| dewey-full | 516.3/5 516.3/520 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/5 516.3/5 20 |
| dewey-search | 516.3/5 516.3/5 20 |
| dewey-sort | 3516.3 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV011196869 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:05:38Z |
| institution | BVB |
| isbn | 0387946802 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007510260 |
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| owner_facet | DE-91G DE-BY-TUM DE-92 DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-703 DE-521 DE-634 DE-83 DE-11 DE-188 |
| physical | XIII, 536 S. graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Springer |
| record_format | marc |
| series2 | Undergraduate texts in mathematics |
| spelling | Cox, David A. Verfasser aut Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra David Cox ; John Little ; Donal O'Shea 2. ed. New York [u.a.] Springer 1997 XIII, 536 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Undergraduate texts in mathematics Literaturverz. S. 523 - 526 Datenverarbeitung Geometry, Algebraic -- Data processing Commutative algebra -- Data processing Datenverarbeitung (DE-588)4011152-0 gnd rswk-swf Algorithmische Geometrie (DE-588)4130267-9 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Computeralgebra (DE-588)4010449-7 gnd rswk-swf Kommutative Algebra (DE-588)4164821-3 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 s Computeralgebra (DE-588)4010449-7 s DE-604 Kommutative Algebra (DE-588)4164821-3 s Datenverarbeitung (DE-588)4011152-0 s 1\p DE-604 Algorithmische Geometrie (DE-588)4130267-9 s 2\p DE-604 Little, John B. Verfasser aut O'Shea, Donal 1952- Verfasser (DE-588)113289731 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007510260&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 | Cox, David A. Little, John B. O'Shea, Donal 1952- Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra Datenverarbeitung Geometry, Algebraic -- Data processing Commutative algebra -- Data processing Datenverarbeitung (DE-588)4011152-0 gnd Algorithmische Geometrie (DE-588)4130267-9 gnd Algebraische Geometrie (DE-588)4001161-6 gnd Computeralgebra (DE-588)4010449-7 gnd Kommutative Algebra (DE-588)4164821-3 gnd |
| subject_GND | (DE-588)4011152-0 (DE-588)4130267-9 (DE-588)4001161-6 (DE-588)4010449-7 (DE-588)4164821-3 |
| title | Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra |
| title_auth | Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra |
| title_exact_search | Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra |
| title_full | Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra David Cox ; John Little ; Donal O'Shea |
| title_fullStr | Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra David Cox ; John Little ; Donal O'Shea |
| title_full_unstemmed | Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra David Cox ; John Little ; Donal O'Shea |
| title_short | Ideals, varieties, and algorithms |
| title_sort | ideals varieties and algorithms an introduction to computational algebraic geometry and commutative algebra |
| title_sub | an introduction to computational algebraic geometry and commutative algebra |
| topic | Datenverarbeitung Geometry, Algebraic -- Data processing Commutative algebra -- Data processing Datenverarbeitung (DE-588)4011152-0 gnd Algorithmische Geometrie (DE-588)4130267-9 gnd Algebraische Geometrie (DE-588)4001161-6 gnd Computeralgebra (DE-588)4010449-7 gnd Kommutative Algebra (DE-588)4164821-3 gnd |
| topic_facet | Datenverarbeitung Geometry, Algebraic -- Data processing Commutative algebra -- Data processing Algorithmische Geometrie Algebraische Geometrie Computeralgebra Kommutative Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007510260&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT coxdavida idealsvarietiesandalgorithmsanintroductiontocomputationalalgebraicgeometryandcommutativealgebra AT littlejohnb idealsvarietiesandalgorithmsanintroductiontocomputationalalgebraicgeometryandcommutativealgebra AT osheadonal idealsvarietiesandalgorithmsanintroductiontocomputationalalgebraicgeometryandcommutativealgebra |