Difference Algebra:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
[New York]
Springer
2008
|
| Schriftenreihe: | Algebra and Applications
8 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 519 S. |
| ISBN: | 9781402069468 |
Internformat
MARC
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| 100 | 1 | |a Levin, Alexander B. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Difference Algebra |c Alexander Levin |
| 264 | 1 | |a [New York] |b Springer |c 2008 | |
| 300 | |a XI, 519 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Algebra and Applications |v 8 | |
| 650 | 0 | 7 | |a Differenzenalgebra |0 (DE-588)4149794-6 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Differenzenalgebra |0 (DE-588)4149794-6 |D s |
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| 830 | 0 | |a Algebra and Applications |v 8 |w (DE-604)BV035420975 |9 8 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016529929 | ||
Datensatz im Suchindex
| _version_ | 1804137701629231104 |
|---|---|
| adam_text | Contents
1
Preliminaries
1
1.1 Basic
Terminology and Background Material
........... 1
1.2
Elements of the Theory of Commutative Rings
.......... 15
1.3
Graded and Filtered Rings and Modules
.............. 37
1.4
Numerical Polynomials
........................ 47
1.5
Dimension Polynomials of Sets of m-tuples
............ 53
1.6
Basic Facts of the Field Theory
................... 64
1.7
Derivations and Modules of Differentials
.............. 89
1.8 Gröbner
Bases
............................ 96
2
Basic Concepts of Difference Algebra
103
2.1
Difference and
Inversive
Difference Rings
.............. 103
2.2
Rings of Difference and
Inversive
Difference Polynomials
..... 115
2.3
Difference Ideals
........................... 121
2.4
Autoreduced Sets of Difference and
Inversive
Difference
Polynomials. Characteristic Sets
.................. 128
2.5 Ritt
Difference Rings
......................... 141
2.6
Varieties of Difference Polynomials
................. 149
3
Difference Modules
155
3.1
Ring of Difference Operators. Difference Modules
......... 155
3.2
Dimension Polynomials of Difference Modules
........... 157
3.3 Gröbner
Bases with Respect to Several Orderings
and
Multivariable
Dimension Polynomials of Difference
Modules
................................ 166
3.4
Inversive
Difference Modules
..................... 185
3.5
¿^-Dimension Polynomials and their Invariants
.......... 195
3.6
Dimension of General Difference Modules
............. 232
ix
Difference Field Extensions
245
4.1 Transformal
Dependence. Difference Transcendental Bases
and Difference Transcendental Degree
............... 245
4.2
Dimension Polynomials of Difference and
Inversive
Difference
Field Extensions
........................... 255
4.3
Limit Degree
............................. 274
4.4
The Fundamental Theorem on Finitely Generated Difference
Field Extensions
........................... 292
4.5
Some Results on Ordinary Difference Field Extensions
...... 295
4.6
Difference Algebras
.......................... 300
Compatibility, Replicability, and Monadicity
311
5.1
Compatible and Incompatible Difference Field Extensions
.... 311
5.2
Difference Kernels over Ordinary Difference Fields
........ 319
5.3
Difference Specializations
...................... 328
5.4
Babbitt s Decomposition. Criterion of Compatibility
....... 332
5.5
Replicability
.............................. 352
5.6
Monadicity
.............................. 354
Difference Kernels over Partial Difference Fields. Difference
Valuation Rings
371
6.1
Difference Kernels over Partial Difference Fields
and their Prolongations
....................... 371
6.2
Realizations of Difference Kernels over Partial Difference Fields
. 376
6.3
Difference Valuation Rings and Extensions of Difference
Specializations
............................ 385
Systems of Algebraic Difference Equations
393
7.1
Solutions of Ordinary Difference Polynomials
........... 393
7.2
Existence Theorem for Ordinary Algebraic Difference
Equations
............................... 402
7.3
Existence of Solutions of Difference Polynomials in the Case
of Two Translations
......................... 412
7.4
Singular and Multiple Realizations
................. 420
7.5
Review of Further Results on Varieties of Ordinary Difference
Polynomials
.............................. 425
7.6
Ritt s Number. Greenspan s and Jacobi s Bounds
......... 433
7.7
Dimension Polynomials and the Strength of a System of Algebraic
Difference Equations
......................... 440
7.8
Computation of Difference Dimension Polynomials in the Case
of Two Translations
......................... 455
Elements of the Difference Galois Theory
463
8.1
Galois Correspondence for Difference Field Extensions
...... 463
8.2
Picard-
Vessiot Theory of Linear Homogeneous Difference
Equations
...............................472
8.3
Picard-
Vessiot
Rings and the Galois Theory of Difference
Equations
...............................486
Bibliography
495
Index
507
|
| adam_txt |
Contents
1
Preliminaries
1
1.1 Basic
Terminology and Background Material
. 1
1.2
Elements of the Theory of Commutative Rings
. 15
1.3
Graded and Filtered Rings and Modules
. 37
1.4
Numerical Polynomials
. 47
1.5
Dimension Polynomials of Sets of m-tuples
. 53
1.6
Basic Facts of the Field Theory
. 64
1.7
Derivations and Modules of Differentials
. 89
1.8 Gröbner
Bases
. 96
2
Basic Concepts of Difference Algebra
103
2.1
Difference and
Inversive
Difference Rings
. 103
2.2
Rings of Difference and
Inversive
Difference Polynomials
. 115
2.3
Difference Ideals
. 121
2.4
Autoreduced Sets of Difference and
Inversive
Difference
Polynomials. Characteristic Sets
. 128
2.5 Ritt
Difference Rings
. 141
2.6
Varieties of Difference Polynomials
. 149
3
Difference Modules
155
3.1
Ring of Difference Operators. Difference Modules
. 155
3.2
Dimension Polynomials of Difference Modules
. 157
3.3 Gröbner
Bases with Respect to Several Orderings
and
Multivariable
Dimension Polynomials of Difference
Modules
. 166
3.4
Inversive
Difference Modules
. 185
3.5
¿^-Dimension Polynomials and their Invariants
. 195
3.6
Dimension of General Difference Modules
. 232
ix
Difference Field Extensions
245
4.1 Transformal
Dependence. Difference Transcendental Bases
and Difference Transcendental Degree
. 245
4.2
Dimension Polynomials of Difference and
Inversive
Difference
Field Extensions
. 255
4.3
Limit Degree
. 274
4.4
The Fundamental Theorem on Finitely Generated Difference
Field Extensions
. 292
4.5
Some Results on Ordinary Difference Field Extensions
. 295
4.6
Difference Algebras
. 300
Compatibility, Replicability, and Monadicity
311
5.1
Compatible and Incompatible Difference Field Extensions
. 311
5.2
Difference Kernels over Ordinary Difference Fields
. 319
5.3
Difference Specializations
. 328
5.4
Babbitt's Decomposition. Criterion of Compatibility
. 332
5.5
Replicability
. 352
5.6
Monadicity
. 354
Difference Kernels over Partial Difference Fields. Difference
Valuation Rings
371
6.1
Difference Kernels over Partial Difference Fields
and their Prolongations
. 371
6.2
Realizations of Difference Kernels over Partial Difference Fields
. 376
6.3
Difference Valuation Rings and Extensions of Difference
Specializations
. 385
Systems of Algebraic Difference Equations
393
7.1
Solutions of Ordinary Difference Polynomials
. 393
7.2
Existence Theorem for Ordinary Algebraic Difference
Equations
. 402
7.3
Existence of Solutions of Difference Polynomials in the Case
of Two Translations
. 412
7.4
Singular and Multiple Realizations
. 420
7.5
Review of Further Results on Varieties of Ordinary Difference
Polynomials
. 425
7.6
Ritt's Number. Greenspan's and Jacobi's Bounds
. 433
7.7
Dimension Polynomials and the Strength of a System of Algebraic
Difference Equations
. 440
7.8
Computation of Difference Dimension Polynomials in the Case
of Two Translations
. 455
Elements of the Difference Galois Theory
463
8.1
Galois Correspondence for Difference Field Extensions
. 463
8.2
Picard-
Vessiot Theory of Linear Homogeneous Difference
Equations
.472
8.3
Picard-
Vessiot
Rings and the Galois Theory of Difference
Equations
.486
Bibliography
495
Index
507 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T21:03:34Z |
| indexdate | 2024-07-09T21:16:29Z |
| institution | BVB |
| isbn | 9781402069468 |
| language | English |
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| physical | XI, 519 S. |
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| series | Algebra and Applications |
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| spelling | Levin, Alexander B. Verfasser aut Difference Algebra Alexander Levin [New York] Springer 2008 XI, 519 S. txt rdacontent n rdamedia nc rdacarrier Algebra and Applications 8 Differenzenalgebra (DE-588)4149794-6 gnd rswk-swf Differenzenalgebra (DE-588)4149794-6 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4020-6947-5 Algebra and Applications 8 (DE-604)BV035420975 8 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016529929&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Levin, Alexander B. Difference Algebra Algebra and Applications Differenzenalgebra (DE-588)4149794-6 gnd |
| subject_GND | (DE-588)4149794-6 |
| title | Difference Algebra |
| title_auth | Difference Algebra |
| title_exact_search | Difference Algebra |
| title_exact_search_txtP | Difference Algebra |
| title_full | Difference Algebra Alexander Levin |
| title_fullStr | Difference Algebra Alexander Levin |
| title_full_unstemmed | Difference Algebra Alexander Levin |
| title_short | Difference Algebra |
| title_sort | difference algebra |
| topic | Differenzenalgebra (DE-588)4149794-6 gnd |
| topic_facet | Differenzenalgebra |
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