Verfügbarkeit: Ideals, varieties, and algorithms | Technische Universität München

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Bibliographische Detailangaben
Beteiligte Personen:	Cox, David A. 1948- (VerfasserIn), Little, John B. 1956- (VerfasserIn), O'Shea, Donal 1952- (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	New York, NY Springer 2012
Ausgabe:	3. ed., corr. at 3. print.
Schriftenreihe:	Undergraduate texts in mathematics
Schlagwörter:	Datenverarbeitung Commutative algebra > Data processing Geometry, Algebraic > Data processing Algorithmische Geometrie Computeralgebra Algebraische Geometrie Kommutative Algebra
Links:	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
Umfang:	XV, 551 S. Ill., graph. Darst.
ISBN:	9780387356501 9781441922571

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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
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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
work_keys_str_mv	AT coxdavida idealsvarietiesandalgorithmsanintroductiontocomputationalalgebraicgeometryandcommutativealgebra AT littlejohnb idealsvarietiesandalgorithmsanintroductiontocomputationalalgebraicgeometryandcommutativealgebra AT osheadonal idealsvarietiesandalgorithmsanintroductiontocomputationalalgebraicgeometryandcommutativealgebra

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