Difference algebra

Alexander Levin1402069464, 9781402069468

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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Table of contents :
Preface……Page 5
Contents……Page 9
1.1 Basic Terminology and Background Material……Page 12
1.2 Elements of the Theory of Commutative Rings……Page 26
1.3 Graded and Filtered Rings and Modules……Page 48
1.4 Numerical Polynomials……Page 58
-tuples……Page 64
1.6 Basic Facts of the Field Theory……Page 75
1.7 Derivations and Modules of Di.erentials……Page 100
1.8 Gr¨obner Bases……Page 107
2.1 Di.erence and Inversive Di.erence Rings……Page 114
2.2 Rings of Di.erence and Inversive Di.erence Polynomials……Page 126
2.3 Di.erence Ideals……Page 132
2.4 Autoreduced Sets of Di.erence and Inversive Di.erence Polynomials. Characteristic Sets……Page 139
2.5 Ritt Di.erence Rings……Page 152
2.6 Varieties of Di.erence Polynomials……Page 160
3.1 Ring of Di.erence Operators. Di.erence Modules……Page 166
3.2 Dimension Polynomials of Di.erence Modules……Page 168
3.3 Gr¨obner Bases with Respect to Several Orderings and Multivariable Dimension Polynomials of Di.erence Modules……Page 177
3.4 Inversive Di.erence Modules……Page 196
-Dimension Polynomials and their Invariants……Page 206
3.6 Dimension of General Di.erence Modules……Page 243
4.1 Transformal Dependence. Di.erence Transcendental Bases and Di.erence Transcendental Degree……Page 256
4.2 Dimension Polynomials of Di.erence and Inversive Di.erence Field Extensions……Page 266
4.3 Limit Degree……Page 285
4.4 The Fundamental Theorem on Finitely Generated Di.erence Field Extensions……Page 303
4.5 Some Results on Ordinary Di.erence Field Extensions……Page 306
4.6 Di.erence Algebras……Page 311
5.1 Compatible and Incompatible Di.erence Field Extensions……Page 321
5.2 Di.erence Kernels over Ordinary Di.erence Fields……Page 329
5.3 Di.erence Specializations……Page 338
5.4 Babbitt’s Decomposition. Criterion of Compatibility……Page 342
5.5 Replicability……Page 362
5.6 Monadicity……Page 364
6.1 Di.erence Kernels over Partial Di.erence Fields and their Prolongations……Page 381
6.2 Realizations of Di.erence Kernels over Partial Di.erence Fields……Page 386
6.3 Di.erence Valuation Rings and Extensions of Di.erence Specializations……Page 395
7.1 Solutions of Ordinary Di.erence Polynomials……Page 402
7.2 Existence Theorem for Ordinary Algebraic Di.erence Equations……Page 411
7.3 Existence of Solutions of Di.erence Polynomials in the Case of Two Translations……Page 421
7.4 Singular and Multiple Realizations……Page 429
7.5 Review of Further Results on Varieties of Ordinary Di.erence Polynomials……Page 434
7.6 Ritt’s Number. Greenspan’s and Jacobi’s Bounds……Page 442
7.7 Dimension Polynomials and the Strength of a System of Algebraic Di.erence Equations……Page 449
7.8 Computation of Di.erence Dimension Polynomials in the Case of Two Translations……Page 464
8.1 Galois Correspondence for Di.erence Field Extensions……Page 472
8.2 Picard-Vessiot Theory of Linear Homogeneous Di.erence Equations……Page 481
8.3 Picard-Vessiot Rings and the Galois Theory of Di.erence Equations……Page 495
Bibliography……Page 503
Index……Page 515