ISBN: 354067733X
TITLE: Computational Commutative Algebra I
AUTHOR: Kreuzer, Martin; Robbiano, Lorenzo
TOC:

Foreword
Introduction 1
0.1 What Is This Book About? 1
0.2 What Is a Gröbner Basis? 2
0.3 Who Invented This Theory? 3
0.4 Now, What Is This Book Really About?. 4
0.5 What Is This Book Not About? 7
0.6 Are There any Applications of This Theory? 8
0.7 How Was This Book Written? 10
0.8 What Is a Tutorial? 11
0.9 What Is CoCoA? 12
0.10 And What Is This Book Good for? 12
0.11 Some Final Words of Wisdom 13
1. Foundations 15
1.1 Polynomial Rings 17
Tutorial 1. Polynomial Representation I 24
Tutorial 2. The Extended Euclidean Algorithm 26
Tutorial 3. Finite Fields 27
1.2 Unique Factorization 29
Tutorial 4. Euclidean Domains 35
Tutorial 5. Squarefree Parts of Polynomials 37
Tutorial 6. Berlekamp's Algorithm 38
1.3 Monomial Ideals and Monomial Modules 41
Tutorial 7. Cogenerators 47
Tutorial 8. Basic Operations with Monomial Ideals and Modules 48
1.4 Term Orderings 49
Tutorial 9. Monoid Orderings Represented by Matrices 57
Tutorial 10. Classification of Term Orderings 58
1.5 Leading Terms. 59
Tutorial 11. Polynomial Representation II 65
Tutorial 12. Symmetric Polynomials 66
Tutorial 13. Newton Polytopes 67
1.6 The Division Algorithm 69
Tutorial 14. Implementation of the Division Algorithm 73
Tutorial 15. Normal Remainders 75
1.7 Gradings 76
Tutorial 16. Homogeneous Polynomials 83
2. Gröbner Bases 85
2.1 Special Generation 87
Tutorial 17. Minimal Polynomials of Algebraic Numbers 89
2.2 Rewrite Rules 91
Tutorial 18. Algebraic Numbers 97
2.3 Syzygies 99
Tutorial 19. Syzygies of Elements of Monomial Modules 108
Tutorial 20. Lifting of Syzygies 108
2.4 Gröbner Bases of Ideals and Modules 110
2.4.A Existence of Gröbner Bases 111
2.4.B Normal Forms 113
2.4.C Reduced Gröbner Bases 115
Tutorial 21. Linear Algebra 119
Tutorial 22. Reduced Gröbner Bases 119
2.5 Buchberger's Algorithm 121
Tutorial 23. Buchberger's Criterion 127
Tutorial 24. Computing Some Gröbner Bases 129
Tutorial 25. Some Optimizations of Buchberger's Algorithm 130
2.6 Hilbert's Nullstellensatz 133
2.6.A The Field-Theoretic Version 134
2.6.B The Geometric Version 137
Tutorial 26. Graph Colourings 143
Tutorial 27. Affine Varieties 143
3. First Applications 145
3.1 Computation of Syzygy Modules 148
Tutorial 28. Splines 155
Tutorial 29. Hilbert's Syzygy Theorem 159
3.2 Elementary Operations on Modules 160
3.2.A Intersections 162
3.2.B Colon Ideals and Annihilators 166
3.2.C Colon Modules 169
Tutorial 30. Computation of Intersections 174
Tutorial 31. Computation of Colon Ideals and Colon Modules 175
3.3 Homomorphisms of Modules 177
3.3.A Kernels, Images, and Liftings of Linear Maps 178
3.3.B Hom-Modules 181
Tutorial 32. Computing Kernels and Pullbacks 191
Tutorial 33. The Depth of a Module 192
3.4 Elimination 195
Tutorial 34. Elimination of Module Components 202
Tutorial 35. Projective Spaces and Gra¡mannians 204
Tutorial 36. Diophantine Systems and Integer Programming 207
3.5 Localization and Saturation 211
3.5.A Localization 212
3.5.B Saturation 215
Tutorial 37. Computation of Saturations 220
Tutorial 38. Toric Ideals 221
3.6 Homomorphisms of Algebras 225
Tutorial 39. Projections 234
Tutorial 40. Gröbner Bases and Invariant Theory 236
Tutorial 41. Subalgebras of Function Fields 239
3.7 Systems of Polynomial Equations 241
3.7.A A Bound for the Number of Solutions 243
3.7.B Radicals of Zero-Dimensional Ideals 246
3.7.C Solving Systems Effectively 254
Tutorial 42. Strange Polynomials 261
Tutorial 43. Primary Decompositions 263
Tutorial 44. Modern Portfolio Theory 267
A. How to Get Started with CoCoA 275
B. How to Program CoCoA 283
C. A Potpourri of CoCoA Programs 293
D. Hints for Selected Exercises 305
Notation 309
Bibliography 313
Index 315
END
