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Difference Algebra:

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Bibliographic Details
Main Author:	Levin, Alexander B. (Author)
Format:	Book
Language:	English
Published:	[New York] Springer 2008
Series:	Algebra and Applications 8
Subjects:	Differenzenalgebra
Online Access:	Inhaltsverzeichnis
Physical Description:	XI, 519 S.
ISBN:	9781402069468

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MARC


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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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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
url	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
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