## Difference AlgebraDifference 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

1 | |

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 |

495 | |

507 | |