| Preliminaries | p. 1 |
| Basic Terminology and Background Material | p. 1 |
| Elements of the Theory of Commutative Rings | p. 15 |
| Graded and Filtered Rings and Modules | p. 37 |
| Numerical Polynomials | p. 47 |
| Dimension Polynomials of Sets of m-tuples | p. 53 |
| Basic Facts of the Field Theory | p. 64 |
| Derivations and Modules of Differentials | p. 89 |
| Grobner Bases | p. 96 |
| Basic Concepts of Difference Algebra | p. 103 |
| Difference and Inversive Difference Rings | p. 103 |
| Rings of Difference and Inversive Difference Polynomials | p. 115 |
| Difference Ideals | p. 121 |
| Autoreduced Sets of Difference and Inversive Difference Polynomials. Characteristic Sets | p. 128 |
| Ritt Difference Rings | p. 141 |
| Varieties of Difference Polynomials | p. 149 |
| Difference Modules | p. 155 |
| Ring of Difference Operators. Difference Modules | p. 155 |
| Dimension Polynomials of Difference Modules | p. 157 |
| Grobner Bases with Respect to Several Orderings and Multivariable Dimension Polynomials of Difference Modules | p. 166 |
| Inversive Difference Modules | p. 185 |
| [sigma]-Dimension Polynomials and their Invariants | p. 195 |
| Dimension of General Difference Modules | p. 232 |
| Difference Field Extensions | p. 245 |
| Transformal Dependence. Difference Transcendental Bases and Difference Transcendental Degree | p. 245 |
| Dimension Polynomials of Difference and Inversive Difference Field Extensions | p. 255 |
| Limit Degree | p. 274 |
| The Fundamental Theorem on Finitely Generated Difference Field Extensions | p. 292 |
| Some Results on Ordinary Difference Field Extensions | p. 295 |
| Difference Algebras | p. 300 |
| Compatibility, Replicability, and Monadicity | p. 311 |
| Compatible and Incompatible Difference Field Extensions | p. 311 |
| Difference Kernels over Ordinary Difference Fields | p. 319 |
| Difference Specializations | p. 328 |
| Babbitt's Decomposition. Criterion of Compatibility | p. 332 |
| Replicability | p. 352 |
| Monadicity | p. 354 |
| Difference Kernels over Partial Difference Fields. Difference Valuation Rings | p. 371 |
| Difference Kernels over Partial Difference Fields and their Prolongations | p. 371 |
| Realizations of Difference Kernels over Partial Difference Fields | p. 376 |
| Difference Valuation Rings and Extensions of Difference Specializations | p. 385 |
| Systems of Algebraic Difference Equations | p. 393 |
| Solutions of Ordinary Difference Polynomials | p. 393 |
| Existence Theorem for Ordinary Algebraic Difference Equations | p. 402 |
| Existence of Solutions of Difference Polynomials in the Case of Two Translations | p. 412 |
| Singular and Multiple Realizations | p. 420 |
| Review of Further Results on Varieties of Ordinary Difference Polynomials | p. 425 |
| Ritt's Number. Greenspan's and Jacobi's Bounds | p. 433 |
| Dimension Polynomials and the Strength of a System of Algebraic Difference Equations | p. 440 |
| Computation of Difference Dimension Polynomials in the Case of Two Translations | p. 455 |
| Elements of the Difference Galois Theory | p. 463 |
| Galois Correspondence for Difference Field Extensions | p. 463 |
| Picard-Vessiot Theory of Linear Homogeneous Difference Equations | p. 472 |
| Picard-Vessiot Rings and the Galois Theory of Difference Equations | p. 486 |
| Bibliography | p. 495 |
| Index | p. 507 |
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