计算可交换的代数 Computational commutative algebra 1
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作者： Martin Kreuzer，Lorenzo Robbiano著
出 版 社：
出版时间： 2008-8-1字数：版次：页数： 321印刷时间： 2008/08/01开本： 16开印次：纸张： 胶版纸I S B N ： 9783540677338包装： 精装内容简介
This is a book about Gröbner bases and their applications. It contains 3 chapters, 20 sections, 44 tutorials, 165 exercises, and numerous further amusements. It is going to help you bridge the gap between theoretical computer algebra and actual computation. We hope you will have as much fun reading it as the authors had writing it!
From the reviews:
"This is one of the most refreshing mathematical books I have ever held in my hands. This is academic teaching at its best;" (H.Stetter, IMN 2003)
"Every paragraph of the book shows how much the authors have enjoyed translating into printed matter the outcome of a long, large, deep and personal relation with computationally oriented commutative algebra." (L.González-Vega and T.Recio, ACM SIGSAM Bull. 2004)
"Each section begins with a quotation and an overview in which "Italian imagination overtakes German rigor". These introductions and the following main bodies of each section are well written, engaging and often amusing. The book is a pleasure to read." (J.Little, Math. Reviews 2001)
目录
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