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| Foreword Introduction 0.1 What Is This Book About? 0.2 What Is a GrSbner Basis? 0.3 Who Invented This Theory? 0.4 Now, What Is This Book Really About? 0.5 What Is This Book Not About? 0.6 Are There any Applications of This Theory? 0.7 How Was This Book Written? 0.8 What Is a Tutorial? 0.9 What Is CoCoA? 0.10 And What Is This Book Good for? 0.11 Some Final Words of Wisdom 1. Foundations 1.1 Polynomial Pdngs Tutorial 1. Polynomial Representation I Tutorial 2. The Extended Euclidean Algorithm Tutorial 3. Finite Fields 1.2 Unique Factorization Tutorial 4. Euclidean Domains Tutorial 5. Squarefree Parts of Polynomials Tutorial 6. Berlekamp's Algorithm 1.3 Monomial Ideals and Monomial Modules Tutorial 7. Cogenerators Tutorial 8. Basic Operations with Monomial Ideals and Modules 1.4 Term Orderings Tutorial 9. Monoid Orderings Represented by Matrices Tutorial 10. Classification of Term Orderings 1.5 Leading Terms Tutorial 11. Polynomial Representation II Tutorial 12. Symmetric Polynomials Tutorial 13. Newton Polytopes 1.6 The Division Algorithm Tutorial 14. Implementation of the Division Algorithm Tutorial 15. Normal Remainders 1.7 Gradings Tutorial 16. Homogeneous Polynomials 2. GrSbner Bases 2.1 Special Generation Tutorial 17. Minimal Polynomials of Algebraic Numbers 2.2 Rewrite Rules Tutorial 18. Algebraic Numbers 2.3 Syzygies Tutorial 19. Syzygies of Elements of Monomial Modules. Tutorial 20. Lifting of Syzygies 2.4 GrSbner Bases of Ideals and Modules 2.4.A Existence of GrSbner Bases 2.4.B Normal Forms 2.4.C Reduced GrSbner Bases Tutorial 21. Linear Algebra Tutorial 22. Reduced GrSbner Bases 2.5 Buchberger's Algorithm Tutorial 23. Buchberger's Criterion Tutorial 24. Computing Some GrSbner Bases Tutorial 25. Some Optimizations of Buchberger's Algorithm 2.6 Hilbert's Nullstellensatz 2.6.A The Field-Theoretic Version 2.6.B The Geometric Version Tutorial 26. Graph Colourings Tutorial 27. Affine Varieties 3. First Applications 3.1 Computation of Syzygy Modules Tutorial 28. Splines Tutorial 29. Hilbert's Syzygy Theorem 3.2 Elementary Operations on Modules 3.2.A Intersections 3.2.B Colon Ideals and Annihilators 3.2.C Colon Modules Tutorial 30. Computation of Intersections Tutorial 31. Computation of Colon Ideals and Colon Modules 3.3 Homomorphisms of Modules 3.3.A Kernels, Images, and Liftings of Linear Maps 3.3.B Horn-Modules Tutorial 32. Computing Kernels and Pullbacks Tutorial 33. The Depth of a Module …… A.How to Get Started with CoCoA B.How to Program CoCoA C.A Potpourri of CoCoA Programs D.Hints for Selected Exercises Notation Bibliography Index |
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