Polyteknisk Boghandel

The main topic of this book is that of Groebner bases & their applications. The main purpose of this book is that of bridging the current gap in the literature between theory & real computation. The book can be used by teachers & students alike as a comprehensive guide to both the theory & the practice of "Computational Commutative Algebra." It has been made as self-contained as possible, & thus is ideally suited as a textbook for graduate or advanced undergraduate courses. Numerous applications are described, covering fields as disparate as algebraic geometry & financial markets. To aid a deeper understanding of these applications there are 44 tutorials aimed at illustrating how the theory can be used in these cases. The computational aspects of the tutorials can be carried out with the computer algebra system CoCoA, an introduction to which appears in an appendix. Besides the tutorials there are plenty of exercises, some of a theoretical nature & others more practical.

Foreword
Introduction
0.
1 What Is This Book About?
0.
2 What Is a Groebner 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
Chapter 1. Foundations
1.
1 Polynomial Rings 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. Berlekamps 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
Chapter 2. Grbner 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. Computing Syzygies of Monomial Modules Tutorial
20. Lifting of Syzygies
2.
4 Grbner Bases of Ideals and Modules
2.
4. A Existence of Grbner Bases
2.
4. B Normal Forms
2.
4. C Reduced Grbner Bases Tutorial
21. Linear Algebra Tutorial
22. Reduced Grbner Bases
2.
5 Buchbergers Algorithm Tutorial
23. Buchbergers Criterion Tutorial
24. Computing Some Grbner Bases Tutorial
25. Some Optimizations of Buchbergers Algorithm
2.
6 Hilberts Nullstellensatz
2.
6. A The Field-Theoretic Version
2.
6. B The Geometric Version Tutorial
26. Graph Colourings Tutorial
27. Affine Varieties
Chapter 3. First Applications
3.
1 Computation of Syzygy Modules Tutorial
28. Splines Tutorial
29. Hilberts 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 Hom-Modules Tutorial
32. Computing Kernels and Pullbacks Tutorial
33. The Depth of a Module
3.
4 Elimination Tutorial
34. Elimination of Module Components Tutorial
35. Projective Spaces and Graomannians Tutorial
36. Diophantine Systems and Integer Programming
3.
5 Localization and Saturation
3.
5. A Localization
3.
5. B Saturation Tutorial
37. Computation of Saturations Tutorial
38. Toric Ideals
3.
6 Homomorphisms of Algebras Tutorial
39. Projections Tutorial
40. Grbner Bases and Invariant Theory Tutorial
41. Subalgebras of Function Fields
3.
7 Systems of Polynomial Equations
3.
7. A A Bound for the Number of Solutions
3.
7. B Radicals of Zero-Dimensional Ideals
3.
7. C Solving Systems Effectively Tutorial
42. Strange Polynomials Tutorial
43. Primary Decompositions Tutorial
44. Modern Portfolio Theory
Appendix A. How to Get Started with CoCoA
Appendix B. How to Program CoCoA
Appendix C. A Potpourri of CoCoA Programs
Appendix D. Hints for Selected Exercises Notation Bibliography
Index