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
2012
|
| Edition: | 3. ed., corr. at 3. print. |
| Series: | Undergraduate texts in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XV, 551 S. Ill., graph. Darst. |
| ISBN: | 9780387356501 9781441922571 |
Staff View
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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. at 3. print. | ||
| 264 | 1 | |a New York, NY |b Springer |c 2012 | |
| 300 | |a XV, 551 S. |b Ill., graph. Darst. | ||
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Record in the Search Index
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| 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
§
I. Introduction
........................ 49
§2.
Orderings
on the Monomials in k[xi
,...,
xn]
........... 54
§3.
A Division Algorithm in k[x
ι,
. ..,.*„]............. 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
ХШ
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.
Zari
ski Closure and Quotients of Ideals
............. 193
§5.
Irreducible Varieties and PrimeIdeals
.............. 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
§
L
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.
Protective Algebraic Geometry
357
§
L
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
§
.
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
С
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
Algebraic Geometry is the study of systems of polynomial equations in one or more variables,
asking such questions as: Does the system have finitely many solutions, and if so how can one
find them? And if there are infinitely many solutions, how can they be described and manip¬
ulated?
The solutions of a system of polynomial equations form a geometric object called a variety;
the corresponding algebraic object is an ideal. There is a close relationship between ideals and
varieties which reveals the intimate link between algebra and geometry. Written at a level
appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the
Nullstellensatz,
invariant
theory, projective geometry, and dimension theory,
The algorithms to answer questions such as those posed above are
ön
important part
of olge*
braic geometry. Although the algorithmic roots of algebraic geometry ore
oli
¡í
îs
only in the
last forty years that computational methods have regained their eorlier prominence. New
algorithms, coupled with the power of fast computers, hove led to
botti
theoretical advances
and interesting applications, for example in robotics and in geometric theorem proving.
In addition to enhancing the text of the second edition, with over
200
pages reflecting changes
to enhance clarity and correctness, this third edition of
¡kok, Vorieïies
anáÁlgof ém
includes:
•
A significantly updated section on Maple in Appendix
С
•
Updated information on AXIOM, (oCoA, Mocaulay
2,
Magma,
Mathematica
and SINGULAR
•
A shorter proof of the Extension Theorem presented in Section
6
of Chapter
3
From the 2nd Edition:
Ί
consider the book to be wonderful.
...
The exposition is very clear, there are many helpful
pictures, and there are a great many instructive exercises, some quite challenging
...
offers
the heart and soul of modern commutative and
algebrák
geometry,
-
fte Amricon
fÁQíkmíkoJ
Monthly
|
| any_adam_object | 1 |
| author | Cox, David A. 1948- Little, John B. 1956- O'Shea, Donal 1952- |
| author_GND | (DE-588)137410832 (DE-588)137410859 (DE-588)113289731 |
| author_facet | Cox, David A. 1948- Little, John B. 1956- O'Shea, Donal 1952- |
| author_role | aut aut aut |
| author_sort | Cox, David A. 1948- |
| author_variant | d a c da dac j b l jb jbl d o do |
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| dewey-full | 510 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
| dewey-raw | 510 |
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| discipline | Mathematik |
| edition | 3. ed., corr. at 3. print. |
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| id | DE-604.BV041107155 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:39:47Z |
| institution | BVB |
| isbn | 9780387356501 9781441922571 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026083438 |
| oclc_num | 854742844 |
| open_access_boolean | |
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| owner_facet | DE-19 DE-BY-UBM DE-706 DE-83 DE-898 DE-BY-UBR DE-355 DE-BY-UBR |
| physical | XV, 551 S. Ill., graph. Darst. |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Springer |
| record_format | marc |
| series2 | Undergraduate texts in mathematics |
| spelling | Cox, David A. 1948- Verfasser (DE-588)137410832 aut Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra David Cox ; John Little ; Donal O'Shea 3. ed., corr. at 3. print. New York, NY Springer 2012 XV, 551 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Undergraduate texts in mathematics Datenverarbeitung Commutative algebra Data processing Geometry, Algebraic Data processing Algorithmische Geometrie (DE-588)4130267-9 gnd rswk-swf Computeralgebra (DE-588)4010449-7 gnd rswk-swf Datenverarbeitung (DE-588)4011152-0 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Kommutative Algebra (DE-588)4164821-3 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. 1956- Verfasser (DE-588)137410859 aut O'Shea, Donal 1952- Verfasser (DE-588)113289731 aut Erscheint auch als Online-Ausgabe 978-0-387-35651-8 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026083438&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026083438&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 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. 1948- Little, John B. 1956- O'Shea, Donal 1952- Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra Datenverarbeitung Commutative algebra Data processing Geometry, Algebraic Data processing Algorithmische Geometrie (DE-588)4130267-9 gnd Computeralgebra (DE-588)4010449-7 gnd Datenverarbeitung (DE-588)4011152-0 gnd Algebraische Geometrie (DE-588)4001161-6 gnd Kommutative Algebra (DE-588)4164821-3 gnd |
| subject_GND | (DE-588)4130267-9 (DE-588)4010449-7 (DE-588)4011152-0 (DE-588)4001161-6 (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 Commutative algebra Data processing Geometry, Algebraic Data processing Algorithmische Geometrie (DE-588)4130267-9 gnd Computeralgebra (DE-588)4010449-7 gnd Datenverarbeitung (DE-588)4011152-0 gnd Algebraische Geometrie (DE-588)4001161-6 gnd Kommutative Algebra (DE-588)4164821-3 gnd |
| topic_facet | Datenverarbeitung Commutative algebra Data processing Geometry, Algebraic Data processing Algorithmische Geometrie Computeralgebra Algebraische Geometrie Kommutative Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026083438&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026083438&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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