1.1 Basic Terminology and BackgroundMaterial . . . . . . . . . . . 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

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

3.1 Ring of Difference Operators. Difference Modules . . . . . . . . . 155

3.2 Dimension Polynomials of Difference Modules . . . . . . . . . . . 157

3.3 Gr¨obner Bases with Respect to Several Orderings

and Multivariable Dimension Polynomials of Difference

Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

3.4 Inversive DifferenceModules . . . . . . . . . . . . . . . . . . . . . 185

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

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

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

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

8.1 Galois Correspondence for Difference Field Extensions . . . . . . 463

8.2 Picard-Vessiot Theory of Linear Homogeneous Difference

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472