Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2008
|
| Edition: | 3. ed., (corr. as of the 2. printing) |
| Series: | Undergraduate texts in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 551 S. Ill., graph. Darst. |
| ISBN: | 0387356509 9780387356501 9780387356518 |
Staff View
MARC
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| 245 | 1 | 0 | |a Ideals, varieties, and algorithms |b an introduction to computational algebraic geometry and commutative algebra |c David Cox ; John Little ; Donal O'Shea |
| 250 | |a 3. ed., (corr. as of the 2. printing) | ||
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Record in the Search Index
| _version_ | 1804138065638195200 |
|---|---|
| adam_text | Contents
Preface
to the First Edition
vii
Preface to the Second Edition
ix
Preface to the Third Edition
xi
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
................. 38
2.
Groebner Bases
49
§ 1.
Introduction
........................ 49
§2.
Orderings
on the Monomials in k[x ,
...
,xn]
........... 54
§3.
A Division Algorithm in £[jq,
...
,xn]
............. 61
§4.
Monomial Ideals and Dickson s Lemma
............. 69
§5.
The Hubert Basis Theorem and Groebner Bases
.......... 75
§6.
Properties of Groebner Bases
................. 82
§7.
Buchberger s Algorithm
................... 88
§8.
First Applications of Groebner Bases
.............. 95
§9.
(Optional) Improvements on Buchberger s Algorithm
....... 102
3.
Elimination Theory
115
§ 1.
The Elimination and Extension Theorems
............ 115
§2.
The Geometry of Elimination
................. 123
§3.
Implicitization
....................... 128
§4.
Singular Points and Envelopes
................. 137
§5.
Unique Factorization and Resultants
.............. 150
§6.
Resultants and the Extension Theorem
............. 162
xiv Contents
4.
The Algebra-Geometry Dictionary
169
§1.
Hubert s
Nullstellensatz................... 169
§2.
Radical Ideals and the Ideal-Variety Correspondence
........ 175
§3.
Sums, Products, and Intersections of Ideals
............ 183
§4.
Zariski Closure and Quotients of Ideals
............. 193
§5.
Irreducible Varieties and Prime Ideals
.............. 198
§6.
Decomposition of a Variety into
Irreducibles
........... 204
§7.
(Optional) Primary Decomposition of Ideals
........... 210
§8.
Summary
......................... 214
5.
Polynomial and Rational Functions on a Variety
215
§ 1.
Polynomial Mappings
.................... 215
§2.
Quotients of Polynomial Rings
................ 221
§3.
Algorithmic Computations in k[x ,
...
,xn]/I
.......... 230
§4.
The Coordinate Ring of an
Affine
Variety
............ 239
§5.
Rational Functions on a Variety
................ 248
§6.
(Optional) Proof of the Closure Theorem
............ 258
6.
Robotics and Automatic Geometric Theorem Proving
265
§ 1.
Geometric Description of Robots
............... 265
§2.
The Forward Kinematic Problem
................ 271
§3.
The Inverse Kinematic Problem and Motion Planning
....... 279
§4.
Automatic Geometric Theorem Proving
............. 291
§5.
Wu s Method
....................... 307
7.
Invariant Theory of Finite Groups
317
§ 1.
Symmetric Polynomials
................... 317
§2.
Finite Matrix Groups and Rings of Invariants
........... 327
§3.
Generators for the Ring of Invariants
.............. 336
§4.
Relations Among Generators and the Geometry of Orbits
...... 345
8.
Projective
Algebraic Geometry
357
§1.
The
Projective
Plane
.................... 357
§2.
Projective
Space and
Projective
Varieties
............ 368
§3.
The
Projective
Algebra-Geometry Dictionary
.......... 379
§4.
The
Projective
Closure of an
Affine
Variety
........... 386
§5.
Projective
Elimination Theory
................. 393
§6.
The Geometry of Quadric Hypersurfaces
............ 408
§7.
Bezout s Theorem
..................... 422
9.
The Dimension of a Variety
439
§ 1.
The Variety of a Monomial Ideal
................ 439
§2.
The Complement of a Monomial Ideal
............. 443
Contents xv
§3.
The Hubert Function and the
Dimension
of a Variety
........ 456
§4.
Elementary Properties of Dimension
.............. 468
§5.
Dimension and Algebraic Independence
............. 477
§6.
Dimension and Nonsingularity
................ 484
§7.
The Tangent Cone
..................... 495
Appendix A. Some Concepts from Algebra
509
§1.
Fields and Rings
...................... 509
§2.
Groups
.......................... 510
§3.
Determinants
....................... 511
Appendix B. Pseudocode
513
§ 1.
Inputs, Outputs, Variables, and Constants
............ 513
§2.
Assignment Statements
................... 514
§3.
Looping Structures
..................... 514
§4.
Branching Structures
.................... 515
Appendix C. Computer Algebra Systems
517
§1.
AXIOM
......................... 517
§2.
Maple
.......................... 520
§3.
Mathematica
........................ 522
§4.
REDUCE
......................... 524
§5.
Other Systems
....................... 528
Appendix D. Independent Projects
530
§1.
General Comments
..................... 530
§2.
Suggested Projects
..................... 530
References
535
Index
541
|
| adam_txt |
Contents
Preface
to the First Edition
vii
Preface to the Second Edition
ix
Preface to the Third Edition
xi
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
. 38
2.
Groebner Bases
49
§ 1.
Introduction
. 49
§2.
Orderings
on the Monomials in k[x\,
.
,xn]
. 54
§3.
A Division Algorithm in £[jq,
.
,xn]
. 61
§4.
Monomial Ideals and Dickson's Lemma
. 69
§5.
The Hubert Basis Theorem and Groebner Bases
. 75
§6.
Properties of Groebner Bases
. 82
§7.
Buchberger's Algorithm
. 88
§8.
First Applications of Groebner Bases
. 95
§9.
(Optional) Improvements on Buchberger's Algorithm
. 102
3.
Elimination Theory
115
§ 1.
The Elimination and Extension Theorems
. 115
§2.
The Geometry of Elimination
. 123
§3.
Implicitization
. 128
§4.
Singular Points and Envelopes
. 137
§5.
Unique Factorization and Resultants
. 150
§6.
Resultants and the Extension Theorem
. 162
xiv Contents
4.
The Algebra-Geometry Dictionary
169
§1.
Hubert's
Nullstellensatz. 169
§2.
Radical Ideals and the Ideal-Variety Correspondence
. 175
§3.
Sums, Products, and Intersections of Ideals
. 183
§4.
Zariski Closure and Quotients of Ideals
. 193
§5.
Irreducible Varieties and Prime Ideals
. 198
§6.
Decomposition of a Variety into
Irreducibles
. 204
§7.
(Optional) Primary Decomposition of Ideals
. 210
§8.
Summary
. 214
5.
Polynomial and Rational Functions on a Variety
215
§ 1.
Polynomial Mappings
. 215
§2.
Quotients of Polynomial Rings
. 221
§3.
Algorithmic Computations in k[x\,
.
,xn]/I
. 230
§4.
The Coordinate Ring of an
Affine
Variety
. 239
§5.
Rational Functions on a Variety
. 248
§6.
(Optional) Proof of the Closure Theorem
. 258
6.
Robotics and Automatic Geometric Theorem Proving
265
§ 1.
Geometric Description of Robots
. 265
§2.
The Forward Kinematic Problem
. 271
§3.
The Inverse Kinematic Problem and Motion Planning
. 279
§4.
Automatic Geometric Theorem Proving
. 291
§5.
Wu's Method
. 307
7.
Invariant Theory of Finite Groups
317
§ 1.
Symmetric Polynomials
. 317
§2.
Finite Matrix Groups and Rings of Invariants
. 327
§3.
Generators for the Ring of Invariants
. 336
§4.
Relations Among Generators and the Geometry of Orbits
. 345
8.
Projective
Algebraic Geometry
357
§1.
The
Projective
Plane
. 357
§2.
Projective
Space and
Projective
Varieties
. 368
§3.
The
Projective
Algebra-Geometry Dictionary
. 379
§4.
The
Projective
Closure of an
Affine
Variety
. 386
§5.
Projective
Elimination Theory
. 393
§6.
The Geometry of Quadric Hypersurfaces
. 408
§7.
Bezout's Theorem
. 422
9.
The Dimension of a Variety
439
§ 1.
The Variety of a Monomial Ideal
. 439
§2.
The Complement of a Monomial Ideal
. 443
Contents xv
§3.
The Hubert Function and the
Dimension
of a Variety
. 456
§4.
Elementary Properties of Dimension
. 468
§5.
Dimension and Algebraic Independence
. 477
§6.
Dimension and Nonsingularity
. 484
§7.
The Tangent Cone
. 495
Appendix A. Some Concepts from Algebra
509
§1.
Fields and Rings
. 509
§2.
Groups
. 510
§3.
Determinants
. 511
Appendix B. Pseudocode
513
§ 1.
Inputs, Outputs, Variables, and Constants
. 513
§2.
Assignment Statements
. 514
§3.
Looping Structures
. 514
§4.
Branching Structures
. 515
Appendix C. Computer Algebra Systems
517
§1.
AXIOM
. 517
§2.
Maple
. 520
§3.
Mathematica
. 522
§4.
REDUCE
. 524
§5.
Other Systems
. 528
Appendix D. Independent Projects
530
§1.
General Comments
. 530
§2.
Suggested Projects
. 530
References
535
Index
541 |
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| author | Cox, David A. Little, John B. O'Shea, Donal 1952- |
| author_GND | (DE-588)113289731 |
| author_facet | Cox, David A. Little, John B. O'Shea, Donal 1952- |
| author_role | aut aut aut |
| author_sort | Cox, David A. |
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| classification_rvk | SK 240 |
| classification_tum | MAT 140f MAT 130f MAT 535f |
| ctrlnum | (OCoLC)635319521 (DE-599)BVBBV035102668 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 3. ed., (corr. as of the 2. printing) |
| format | Book |
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| id | DE-604.BV035102668 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:14:44Z |
| indexdate | 2024-07-09T21:22:16Z |
| institution | BVB |
| isbn | 0387356509 9780387356501 9780387356518 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016770623 |
| oclc_num | 635319521 |
| open_access_boolean | |
| owner | DE-703 DE-11 DE-91G DE-BY-TUM |
| owner_facet | DE-703 DE-11 DE-91G DE-BY-TUM |
| physical | XV, 551 S. Ill., graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| 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 3. ed., (corr. as of the 2. printing) New York, NY Springer 2008 XV, 551 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Undergraduate texts in mathematics Computeralgebra (DE-588)4010449-7 gnd rswk-swf Datenverarbeitung (DE-588)4011152-0 gnd rswk-swf Kommutative Algebra (DE-588)4164821-3 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Algorithmische Geometrie (DE-588)4130267-9 gnd rswk-swf Kommutative Algebra (DE-588)4164821-3 s Datenverarbeitung (DE-588)4011152-0 s DE-604 Algebraische Geometrie (DE-588)4001161-6 s Algorithmische Geometrie (DE-588)4130267-9 s Computeralgebra (DE-588)4010449-7 s 1\p DE-604 2\p DE-604 Little, John B. Verfasser aut O'Shea, Donal 1952- Verfasser (DE-588)113289731 aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016770623&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 Computeralgebra (DE-588)4010449-7 gnd Datenverarbeitung (DE-588)4011152-0 gnd Kommutative Algebra (DE-588)4164821-3 gnd Algebraische Geometrie (DE-588)4001161-6 gnd Algorithmische Geometrie (DE-588)4130267-9 gnd |
| subject_GND | (DE-588)4010449-7 (DE-588)4011152-0 (DE-588)4164821-3 (DE-588)4001161-6 (DE-588)4130267-9 |
| 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_exact_search_txtP | 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 | Computeralgebra (DE-588)4010449-7 gnd Datenverarbeitung (DE-588)4011152-0 gnd Kommutative Algebra (DE-588)4164821-3 gnd Algebraische Geometrie (DE-588)4001161-6 gnd Algorithmische Geometrie (DE-588)4130267-9 gnd |
| topic_facet | Computeralgebra Datenverarbeitung Kommutative Algebra Algebraische Geometrie Algorithmische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016770623&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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