Titel: Computational Commutative Algebra 1
Autor/en: Martin Kreuzer, Lorenzo Robbiano
1st ed. 2000. Corr. 2nd printing 2008.
Mit Abbildungen und Tabellen.
1. September 2000 - gebunden - X
This introduction to polynomial rings, Gröbner bases and applications bridges the gap in the literature between theory and actual computation. It details numerous applications, covering fields as disparate as algebraic geometry and financial markets. To aid in a full understanding of these applications, more than 40 tutorials illustrate how the theory can be used. The book also includes many exercises, both theoretical and 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
From the reviews:
"... This is one of the most refreshing mathematical books I have ever held in my hands. The authors do not believe in teaching by spreading out the material, but they introduce it via questions and discussions, they explore it in an intuitive fashion, exercise it through well-chosen examples, and start the reader on his own expeditions through numerous "tutorials", i.e., guided projects. This is academic teaching at its best: if I had not seen it, I should not have believed that it can be done so well. ... In conclusion, this book gives students a stimulating introduction to commutative algebra very much geared to their need, and it provides numerous useful ideas to those who teach the subject." (H.Stetter, IMN - Internationale Mathematische Nachrichten 2003, Vol. 57, Issue 193)
"This account of polynomial rings, Gröbner bases and applications such as computations of syzygies is written from the point of view of a computer user; as a result, it often provides new insights. The exercises and tutorials (how to work with CoCoA) add to the usefulness of the volume." (Mathematika 48, 2001)
"Das Buch ist in einem aufmunternd lockeren Stil geschrieben und für Studierende gut zum Selbststudium geeignet. Der präsentierte Stoff wird durch viele Beispiele und Übungsaufgaben (teilweise mit Anleitungen im Anhang) ergänzt. Weiter sind in jedem Abschnitt sorgfältig ausgearbeitete Tutorials angefügt, die zur Benutzung von Computeralgebrasystemen, insbesondere CoCoA, anregen sollen." (Computeralgebra-Rundbrief, Nr. 28, März 2001)
"Four years ago the authors published the first volume of a projected seris about computational commutative algebra which "took three years of intense work just to fill three centimeters of your bookshelf". Now they have gifted us with the second volume, and they say that "the completion of this volume took four years and it is about four centimeters thick. Thus we have a confirmed invariant which governs our writing: our velocity is one centimeter per year." These quotations from the foreword of the book give a clue of how amusing it is, and also how mathematically solid it is.
In this second volume the authors continue with the same style as the first, an almost humorous one. [...]
It was a pleasure to review this nice book but, as always, the equilibrium of the world depends on good and bad things, and the "bad thing" is the authors' decision to not write a third volume. In their own words: "Alas, we have to inform you that this is absolutely and definitely the second and last volume of the trilogy." Fortunately, these two volumes collect so large an amount of information and inspiration that we can say the authors actually fulfilled our expectations."
Paulo F. Machado, Mathematical Review Clippings 2006h
"An excellent, non-standard, elementary and self-contained introduction to the theory of Gröbner bases and its applications ... . Moreover, the style is very friendly and relaxing ... . As it is now, this book can be used either to introduce the theory of Gröbner bases to students with a basic knowledge of algebra or to provide a first introduction to commutative algebra ... ." (Laureano Gonzélez-Vega and Tomás Recio, ACM SIGSAM Bulletin, Vol. 38 (2), 2004)