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