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1 Introduction 1

1 Polynomials and Ideals 1

2 Monomial Orders and Polynomial Division 6

3 Gr？bner Bases 13

4 Affine Varieties 19

2 Solving Polynomial Equations 26

1 Solving Polynomial Systems by Elimination 26

2 Finite-Dimensional Algebras 37

3 Gr？bner Basis Conversion 49

4 Solving Equations via Eigenvalues and Eigenvectors 56

5 Real Root Location and Isolation 69

3 Resultants 77

1 The Resultant of Two Polynomials 77

2 Multipolynomial Resultants 84

3 Properties of Resultants 95

4 Computing Resultants 102

5 Solving Equations via Resultants 114

6 Solving Equations via Eigenvalues and Eigenvectors 128

4 Computation in Local Rings 137

1 Local Rings 137

2 Multiplicities and Milnor Numbers 145

3 Term Orders and Division in Local Rings 158

4 Standard Bases in Local Rings 174

5 Applications of Standard Bases 180

5 Modules 189

1 Modules over Rings 189

2 Monomial Orders and Gr？bner Bases for Modules 207

3 Computing Syzygies 222

4 Modules over Local Rings 234

6 Free Resolutions 247

1 Presentations and Resolutions of Modules 247

2 Hilbert's Syzygy Theorem 258

3 Graded Resolutions 266

4 Hilbert Polynomials and Geometric Applications 280

7 Polytopes,Resultants,and Equations 305

1 Geometry of Polytopes 305

2 Sparse Resultants 313

3 Toric Varieties 322

4 Minkowski Sums and Mixed Volumes 332

5 Bernstein's Theorem 342

6 Computing Resultants and Solving Equations 357

8 Polyhedral Regions and Polynomials 376

1 Integer Programming 376

2 Integer Programming and Combinatorics 392

3 Multivariate Polynomial Splines 405

4 The Gr？bner Fan of an Ideal 426

5 The Gr？bner Walk 436

9 Algebraic Coding Theory 451

1 Finite Fields 451

2 Error-Correcting Codes 459

3 Cyclic Codes 468

4 Reed-Solomon Decoding Algorithms 480

10 The Berlekamp-Massey-Sakata Decoding Algorithm 494

1 Codes from Order Domains 494

2 The Overall Structure of the BMS Algorithm 508

3 The Details of the BMS Algorithm 522

References 533

Index 547