Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cham
Springer
[2018]
|
| Ausgabe: | fourth edition, corrected publication |
| Schriftenreihe: | Undergraduate texts in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xvi, 646 Seiten Illustrationen |
| ISBN: | 9783319374277 9783319167206 |
Internformat
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 A. Cox, John Little, Donal O'Shea |
| 250 | |a fourth edition, corrected publication | ||
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Datensatz im Suchindex
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| adam_text |
Contents Preface. vii Notation for Sets and Functions. xv Geometry, Algebra, and Algorithms. 1 1 5 14 29 37 Chapter 1. §1 §2 §3 §4 §5 . . . . . Polynomials and Affine Space. Affine Varieties. Parametrizations of Affine Varieties. Ideals . Polynomials of One Variable. Gröbner Bases. 49 . Introduction. 49 . Orderings on the Monomials in fc[%i,. ,xn]. 54 . A Division Algorithm in k[xi,., x„]. 61 . Monomial Ideals and Dickson’s Lemma. 70 . The Hilbert Basis Theorem and Gröbner Bases. 76 . Properties of Gröbner Bases. 83 . Buchberger’s
Algorithm. 90 . First Applications of Gröbner Bases. 97 . Refinements of the Buchberger Criterion. 104 . Improvements on Buchberger’s Algorithm. 109 Chapter 2. §1 §2 §3 §4 §5 §6 §7 §8 §9 §10 Elimination Theory . 121 The Elimination and Extension Theorems. 121 The Geometry of Elimination . 129 Implicitization. 133 Singular Points and Envelopes. 143 Gröbner Bases and the Extension Theorem. 155 Resultants and the Extension Theorem. 161 Chapter 3 §1 §2 §3 §4 §5 §6 . . . . . . xi
xii Contents Chapter 4. The Algebra-Geometry Dictionary. 175 §1 . Hilbert’s Nullstellensatz. 175 §2 . Radical Ideals and the Ideal-Variety Correspondence. 181 §3 . Sums, Products, and Intersections of Ideals. 189 §4 . Zariski Closures, Ideal Quotients, and Saturations . 199 §5 . Irreducible Varieties and Prime Ideals. 206 §6 . Decomposition of a Variety into Irreducibles. 212 §7, Proof of the Closure Theorem. 219 §8 . Primary Decomposition of Ideals . 228 §9 . Summary. 232 Chapter 5. Polynomial and Rational Functions on a Variety.233 § 1. Polynomial Mappings. 233 §2 . Quotients of Polynomial Rings. 240 §3 . Algorithmic Computations in k[x\,. ,Xn\/I. ·. 248 §4 . The Coordinate Ring of an AffineVariety . 257 §5 . Rational Functions on a Variety. 268 §6 . Relative Finiteness and Noether Normalization. 277
Chapter 6. Robotics and Automatic Geometric Theorem Proving. 291 §1 . Geometric Description of Robots . 291 §2 . The Forward Kinematic Problem . 297 §3 . The Inverse Kinematic Problem and Motion Planning. 304 §4 . Automatic Geometric Theorem Proving. 319 §5 . Wu’s Method. 335 Chapter 7. Invariant Theory of Finite Groups. 345 §1 . Symmetric Polynomials. 345 §2 . Finite Matrix Groups and Rings of Invariants. 355 §3 . Generators for the Ring of Invariants . 364 §4 . Relations Among Generators and the Geometry of Orbits. 373 Chapter 8. Projective Algebraic Geometry. 385 § 1. The Projective Plane. 385 §2 . Projective Space and Projective Varieties. 396 §3 . The Projective Algebra-Geometry Dictionary. 406 §4 . The Projective Closure of an Affine Variety. 415 §5 . Projective Elimination
Theory. 422 §6 . The Geometry of Quadric Hypersurfaces. 436 §7 . Bezout’s Theorem. 451 Chapter 9. The Dimension of a Variety .469 § 1. The Variety of a Monomial Ideal. 469 §2 . The Complement of a Monomial Ideal. 473 §3 . The Hilbert Function and the Dimension of a Variety. 486 §4 . Elementary Properties of Dimension. 498
Contents xiii §5 . Dimension and Algebraic Independence . 506 §6 . Dimension and Nonsingularity . 515 §7 . The Tangent Cone. 525 Chapter §1 §2 §3 §4 10.Additional Gröbner Basis Algorithms. 539 . Preliminaries. 539 . Hilbert Driven Buchberger Algorithms. 550 . The F^ Algorithm. 567 . Signature-based Algorithms and F^ . 576 Correction to: Ideals, Varieties, and Algorithms. El Appendix A. Some Concepts from Algebra .593 §1 . Fields and Rings. 593 §2 . Unique Factorization. 594 §3 . Groups. 595 §4 . Determinants. 596 Appendix B. Pseudocode.599 § 1. Inputs, Outputs,
Variables, and Constants. 600 §2 . Assignment Statements. 600 §3 . Looping Structures.600 §4 . Branching Structures. 602 §5 . Output Statements. 602 Appendix C. Computer Algebra Systems. 603 §1 . General Purpose Systems: Maple, Mathematica, Sage. 604 §2 . Special Purpose Programs: CoCoA, Macaulay2, Singular. 611 §3 . Other Systems and Packages. 617 Appendix D. Independent Projects. 619 §1 . General Comments.619 §2 . Suggested Projects .620 References. 627 Index .635 |
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| author_facet | Cox, David A. 1948- Little, John N. O'Shea, Donal 1952- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | fourth edition, corrected publication |
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| index_date | 2024-07-03T16:44:56Z |
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| isbn | 9783319374277 9783319167206 |
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| spelling | Cox, David A. 1948- Verfasser (DE-588)137410832 aut Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra David A. Cox, John Little, Donal O'Shea fourth edition, corrected publication Cham Springer [2018] © 2018 xvi, 646 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Undergraduate texts in mathematics Mathematics Geometry, algebraic Algebra Computer software Logic, Symbolic and mathematical Algebraic Geometry Commutative Rings and Algebras Mathematical Logic and Foundations Mathematical Software Mathematik Datenverarbeitung (DE-588)4011152-0 gnd rswk-swf Algorithmische Geometrie (DE-588)4130267-9 gnd rswk-swf Kommutative Algebra (DE-588)4164821-3 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 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 N. Verfasser (DE-588)143320874 aut O'Shea, Donal 1952- Verfasser (DE-588)113289731 aut Erscheint auch als Online-Ausgabe 978-3-319-16720-6 Erscheint auch als Online-Ausgabe 978-3-319-16721-3 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032583119&sequence=000001&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. 1948- Little, John N. O'Shea, Donal 1952- Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra Mathematics Geometry, algebraic Algebra Computer software Logic, Symbolic and mathematical Algebraic Geometry Commutative Rings and Algebras Mathematical Logic and Foundations Mathematical Software Mathematik Datenverarbeitung (DE-588)4011152-0 gnd Algorithmische Geometrie (DE-588)4130267-9 gnd Kommutative Algebra (DE-588)4164821-3 gnd Algebraische Geometrie (DE-588)4001161-6 gnd Computeralgebra (DE-588)4010449-7 gnd |
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| 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 A. Cox, John Little, Donal O'Shea |
| title_fullStr | Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra David A. Cox, John Little, Donal O'Shea |
| title_full_unstemmed | Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra David A. 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 | Mathematics Geometry, algebraic Algebra Computer software Logic, Symbolic and mathematical Algebraic Geometry Commutative Rings and Algebras Mathematical Logic and Foundations Mathematical Software Mathematik Datenverarbeitung (DE-588)4011152-0 gnd Algorithmische Geometrie (DE-588)4130267-9 gnd Kommutative Algebra (DE-588)4164821-3 gnd Algebraische Geometrie (DE-588)4001161-6 gnd Computeralgebra (DE-588)4010449-7 gnd |
| topic_facet | Mathematics Geometry, algebraic Algebra Computer software Logic, Symbolic and mathematical Algebraic Geometry Commutative Rings and Algebras Mathematical Logic and Foundations Mathematical Software Mathematik Datenverarbeitung Algorithmische Geometrie Kommutative Algebra Algebraische Geometrie Computeralgebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032583119&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT coxdavida idealsvarietiesandalgorithmsanintroductiontocomputationalalgebraicgeometryandcommutativealgebra AT littlejohnn idealsvarietiesandalgorithmsanintroductiontocomputationalalgebraicgeometryandcommutativealgebra AT osheadonal idealsvarietiesandalgorithmsanintroductiontocomputationalalgebraicgeometryandcommutativealgebra |