Difference Algebra (Google eBook)

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Springer Science & Business Media, Apr 19, 2008 - Mathematics - 532 pages
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Difference algebra grew out of the study of algebraic difference equations with coefficients from functional fields. The first stage of this development of the theory is associated with its founder, J.F. Ritt (1893-1951), and R. Cohn, whose book Difference Algebra (1965) remained the only fundamental monograph on the subject for many years. Nowadays, difference algebra has overgrown the frame of the theory of ordinary algebraic difference equations and appears as a rich theory with applications to the study of equations in finite differences, functional equations, differential equations with delay, algebraic structures with operators, group and semigroup rings. The monograph is intended for graduate students and researchers in difference and differential algebra, commutative algebra, ring theory, and algebraic geometry. The book is self-contained; it requires no prerequisites other than the knowledge of basic algebraic concepts and a mathematical maturity of an advanced undergraduate.
  

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Contents

Preliminaries
1
12 Elements of the Theory of Commutative Rings
15
13 Graded and Filtered Rings and Modules
37
14 Numerical Polynomials
47
15 Dimension Polynomials of Sets of mtuples
53
16 Basic Facts of the Field Theory
64
17 Derivations and Modules of Differentials
89
18 Gröbner Bases
96
46 Difference Algebras
300
Compatibility Replicability and Monadicity
311
52 Difference Kernels over Ordinary Difference Fields
319
53 Difference Specializations
328
54 Babbitts Decomposition Criterion of Compatibility
332
55 Replicability
352
56 Monadicity
354
Difference Kernels over Partial Difference Fields Difference Valuation Rings
371

Basic Concepts of Difference Algebra
103
22 Rings of Difference and Inversive Difference Polynomials
115
23 Difference Ideals
121
24 Autoreduced Sets of Difference and Inversive Difference Polynomials Characteristic Sets
128
25 Ritt Difference Rings
141
26 Varieties of Difference Polynomials
149
Difference Modules
155
32 Dimension Polynomials of Difference Modules
157
33 Grobner Bases with Respect to Several Orderings and MuItivariable Dimension Polynomials of Difference Modules
166
34 Inversive Difference Modules
185
35 rDimension Polynomials and their Invariants
195
36 Dimension of General Difference Modules
232
Difference Field Extensions
245
and Inversive Difference Field Extensions
255
43 Limit Degree
274
44 The Fundamental Theorem on Finitely Generated Difference Field Extensions
292
45 Some Results on Ordinary Difference Field Extensions
295
62 Realizations of Difference Kernels over Partial Difference Fields
376
63 Difference Valuation Rings and Extensions of Difference Specializations
385
Systems of Algebraic Difference Equations
393
72 Existence Theorem for Ordinary Algebraic Difference Equations
402
Polynomials in the Case of Two Translations
412
74 Singular and Multiple Realizations
420
75 Review of Further Results on Varieties of Ordinary Difference Polynomials
425
76 Ritts Number Greenspans and Jacobis Bounds
433
77 Dimension Polynomials and the Strength of a System of Algebraic Difference Equations
440
Polynomials in the Case of Two Translations
455
Elements of the Difference Galois Theory
463
Homogeneous Difference Equations
472
83 PicardVessiot Rings and the Galois Theory of Difference Equations
486
Bibliography
495
Index
507
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Page 496 - The model theory of difference fields II: periodic ideals and the trichotomy in all characteristics

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