Gröbner bases in commutative algebra:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, R.I.
American Mathematical Society
2012
|
| Series: | Graduate studies in mathematics
130 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. 157-159) and index |
| Physical Description: | XI, 164 S. graph. Darst. |
| ISBN: | 0821872877 9780821872871 |
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| 100 | 1 | |a Ene, Viviana |d 1960- |e Verfasser |0 (DE-588)141139382 |4 aut | |
| 245 | 1 | 0 | |a Gröbner bases in commutative algebra |c Viviana Ene ; Jürgen Herzog |
| 264 | 1 | |a Providence, R.I. |b American Mathematical Society |c 2012 | |
| 300 | |a XI, 164 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Graduate studies in mathematics |v 130 | |
| 500 | |a Includes bibliographical references (p. 157-159) and index | ||
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| 650 | 4 | |a Commutative algebra | |
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Record in the Search Index
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| adam_text |
Titel: Gröbner bases in commutative algebra
Autor: Ene, Viviana
Jahr: 2012
Contents
Preface ix
Chapter 1. Polynomial rings and ideals 1
§1.1. Polynomial rings 1
1.1.1. Definition of the polynomial ring 1
1.1.2. Some basic properties of polynomial rings 5
§1.2. Ideals 6
1.2.1. Operations on ideals 6
1.2.2. Residue class rings 7
1.2.3. Monomial ideals and Dickson's lemma 8
1.2.4. Operations on monomial ideals 10
Problems 12
Chapter 2. Gröbner bases 15
§2.1. Monomial orders 15
2.1.1. Examples and basic properties of monomial orders 15
2.1.2. Construction of monomial orders 17
§2.2. Initial ideals and Gröbner bases 18
2.2.1. The basic definitions 18
2.2.2. Macaulay's theorem 20
2.2.3. Hilbert's basis theorem 21
§2.3. The division algorithm 22
§2.4. Buchberger's criterion 25
§2.5. Buchberger's algorithm 28
§2.6. Reduced Gröbner bases 29
Problems 30
vi Contents
Chapter 3. First applications 33
§3.1. Elimination of variables 33
3.1.1. Elimination orders 33
3.1.2. The Elimination Theorem 34
§3.2. Applications to operations on ideals 34
3.2.1. Intersection of ideals 34
3.2.2. Ideal quotient 35
3.2.3. Saturation and radical membership 36
3.2.4. ÜT-algebra homomorphisms 37
3.2.5. Homogenization 40
§3.3. Zero dimensional ideals 42
§3.4. Ideals of initial forms 46
Problems 48
Chapter 4. Gröbner bases for modules 51
§4.1. Modules 51
§4.2. Monomial orders and initial modules 53
§4.3. The division algorithm and Buchberger's criterion and
algorithm for modules 56
§4.4. Syzygies 58
4.4.1. How to compute syzygy modules 58
4.4.2. Systems of linear equations over the polynomial ring 63
4.4.3. Schreyer's theorem 66
4.4.4. Graded rings and modules 68
4.4.5. Graded free resolutions 70
4.4.6. Numerical data arising from graded resolutions 73
4.4.7. Z"-graded modules 76
Problems 80
Chapter 5. Gröbner bases of toric ideals 83
§5.1. Semigroup rings and toric ideals 83
§5.2. Gröbner bases of toric ideals 87
§5.3. Simplicial complexes and squarefree monomial ideals 88
§5.4. Normal semigroup rings 91
§5.5. Edge rings associated with bipartite graphs 94
Problems 97
Chapter 6. Selected applications in commutative algebra and
combinatorics 99
§6.1. Koszul algebras 99
Contents vii
§6.2. Sortable sets of monomials 105
§6.3. Generalized Hibi rings 110
§6.4. Gröbner bases for Rees rings 113
6.4.1. The ¿"-exchange property 113
6.4.2. The Rees ring of generalized Hibi ideals 115
§6.5. Determinantal ideals 117
6.5.1. Determinantal ideals and their initial ideals 117
6.5.2. The initial complex of a determinantal ideal 121
§6.6. Sagbi bases and the coordinate ring of Grassmannians 127
6.6.1. Sagbi bases 127
6.6.2. The coordinate ring of Grassmannians 130
§6.7. Binomial edge ideals 135
§6.8. Connectedness of contingency tables 140
6.8.1. Contingency tables and the x2-statistics 140
6.8.2. Random walks 141
6.8.3. Contingency tables of shape 2 x n 144
Problems 152
Bibliography 157
Index 161 |
| any_adam_object | 1 |
| author | Ene, Viviana 1960- Herzog, Jürgen 1941-2024 |
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| classification_rvk | SK 230 |
| classification_tum | MAT 130f MAT 160f |
| ctrlnum | (OCoLC)774931412 (DE-599)BVBBV040368652 |
| dewey-full | 512/.44 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV040368652 |
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| indexdate | 2025-03-07T13:01:05Z |
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| isbn | 0821872877 9780821872871 |
| language | English |
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| physical | XI, 164 S. graph. Darst. |
| publishDate | 2012 |
| publishDateSearch | 2012 |
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| publisher | American Mathematical Society |
| record_format | marc |
| series | Graduate studies in mathematics |
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| spelling | Ene, Viviana 1960- Verfasser (DE-588)141139382 aut Gröbner bases in commutative algebra Viviana Ene ; Jürgen Herzog Providence, R.I. American Mathematical Society 2012 XI, 164 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 130 Includes bibliographical references (p. 157-159) and index Gröbner bases Commutative algebra Gröbner-Basis (DE-588)4276378-2 gnd rswk-swf Kommutative Algebra (DE-588)4164821-3 gnd rswk-swf Gröbner-Basis (DE-588)4276378-2 s Kommutative Algebra (DE-588)4164821-3 s DE-604 Herzog, Jürgen 1941-2024 Verfasser (DE-588)142854115 aut Graduate studies in mathematics 130 (DE-604)BV009739289 130 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025222305&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ene, Viviana 1960- Herzog, Jürgen 1941-2024 Gröbner bases in commutative algebra Graduate studies in mathematics Gröbner bases Commutative algebra Gröbner-Basis (DE-588)4276378-2 gnd Kommutative Algebra (DE-588)4164821-3 gnd |
| subject_GND | (DE-588)4276378-2 (DE-588)4164821-3 |
| title | Gröbner bases in commutative algebra |
| title_auth | Gröbner bases in commutative algebra |
| title_exact_search | Gröbner bases in commutative algebra |
| title_full | Gröbner bases in commutative algebra Viviana Ene ; Jürgen Herzog |
| title_fullStr | Gröbner bases in commutative algebra Viviana Ene ; Jürgen Herzog |
| title_full_unstemmed | Gröbner bases in commutative algebra Viviana Ene ; Jürgen Herzog |
| title_short | Gröbner bases in commutative algebra |
| title_sort | grobner bases in commutative algebra |
| topic | Gröbner bases Commutative algebra Gröbner-Basis (DE-588)4276378-2 gnd Kommutative Algebra (DE-588)4164821-3 gnd |
| topic_facet | Gröbner bases Commutative algebra Gröbner-Basis Kommutative Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025222305&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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