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Singular Introduction to Commutative Algebra by G. M. Greuel and Gerhard. Pfister (2002, Mixed Lot)
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About this product
Product Identifiers
PublisherSpringer
ISBN-103540428976
ISBN-139783540428978
eBay Product ID (ePID)2220002
Product Key Features
Number of PagesXviii, 588 Pages
LanguageEnglish
Publication NameSingular Introduction to Commutative Algebra
Publication Year2002
SubjectComputer Science, Algebra / General, Geometry / Algebraic
TypeTextbook
AuthorG. M. Greuel, Gerhard. Pfister
Subject AreaMathematics, Computers
FormatMixed Lot
Dimensions
Item Weight33.1 Oz
Item Length9.3 in
Item Width6.1 in
Additional Product Features
Intended AudienceCollege Audience
LCCN2002-070475
Reviews"a? It is certainly no exaggeration to say that Greuel and Pfister's A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra, in which computational methods and results become central to how the subject is taught and learned. [a? ] Among the great strengths and most distinctive features of Greuel and Pfister's book is a new, completely unified treatment of the global and local theories. The realization that the two cases could be combined to this extent was decisive in the design of the Singular system, making it one of the most flexible and most efficient systems of its type. The authors present the first systematic development of this unified approach in a textbook here, and this aspect alone is almost worth the price of admission. Another distinctive feature of this book is the degree of integration of explicit computational examples into the flow of the text. Strictly mathematical components of the development (often quite terse and written in a formal "theorem-proof" style) are interspersed with parallel discussions of features of Singular and numerous Singular examples giving input commands, some extended programs in the Singular language, and output. [a? ] Yet another strength of Greuel and Pfister's book is its breadth of coverage of theoretical topics in the portions of commutative algebra closest to algebraic geometry, with algorithmic treatments of almost every topic. A synopsis of the table of contents will make this clear. [a? ] Greuel and Pfister have written a distinctive an highly useful book that should be in the library of every commutative algebrais and algebraic geometer, expert and novice alike. I hope that it achieves the educational impact it deserves." John B. Little, Monthly of The Mathematical Association of America, March 2004 "... The authors' most important new focus is the presentation of non-well orderings that allow them the computational approach for local commutative algebra. The accompanying CD-ROM also contains all the examples of the book. ... In fact the book provides an introduction to commutative algebra from a computational point of view. So it might be helpful for students and other interested readers (familiar with computers) to explore the beauties and difficulties of commutative algebra by computational experiences. In this respect the book is the one of the first samples of a new kind of textbooks in algebra." P.Schenzel, Zentralblatt f'r Mathematik 1023.13001, 2003 "It is certainly no exaggeration to say that a? A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra a? . Among the great strengths and most distinctive features a? is a new, completely unified treatment of the global and local theories. a? Greuel and Pfister have written a distinctive and highly useful book that should be in the library of every commutative algebraist and algebraic geometer, expert and novice alike." (John B. Little, MAA, March 2004) "The aim of the book is a? an introduction to commutative algebra with a view towards to algorithmic aspects and computational practice. a? The authorsa? most important new focus is the presentation of non-well orderings that allow them the computational approach for local commutative algebra. a? It might be helpful for students and other interested readers a? to explore the beauties and difficulties of commutative algebra a? . The book is one of the first samples of a new kind of textbooks in algebra." (Peter Schenzel, Zentralblatt MATH, Vol. 1023, 2003)
TitleLeadingA
Dewey Edition21
IllustratedYes
Dewey Decimal512/.24
Table Of Content1 Rings, Ideals and Standard Bases 1.1 Rings, Polynomials and Ring Maps 1.2 Monomial Orderings 1.3 Ideals and Quotient Rings 1.4 Local Rings and Localization 1.5 Rings Associated to Monomial Orderings 1.6 Normal Forms and Standard Bases 1.7 The Standard Basis Algorithm 1.8 Operations on Ideals and their Computation 1.8.1 Ideal membership 1.8.2 Intersection with subrings (elimination of variables) 1.8.3 Zariski closure of the image 1.8.4 Solvability of polynomial equations 1.8.5 Solving polynomial equations 1.8.6 Radical membership 1.8.7 Intersection of ideals 1.8.8 Quotient of ideals 1.8.9 Saturation 1.8.10 Kernel of a ring map 1.8.11 Algebraic dependence and subalgebra membership2. Modules 2.1 Modules, Submodules and Homomorphisms 2.2 Graded Rings and Modules 2.3 Standard Bases for Modules 2.4 Exact Sequences and free Resolutions 2.5 Computing Resolutions and the Syzygy Theorem 2.6 Modules over Principal Ideal Domains 2.7 Tensor Product 2.8 Operations with modules 2.8.1 Module membership problem 2.8.2 Elimination of module components 2.8.3 Quotients of submodules 2.8.4 Kernel of a module homomorphism3. Noether Normalization and Applications 3.1 Finite and Integral Extensions 3.2 The Integral Closure 3.3 Dimension 3.4 Noether Normalization 3.5 Applications 3.6 An Algorithm to Compute the Normalization 3.7 Procedures4. Primary Decomposition and Related Topics 4.1 The Theory of Primary Decomposition 4.2 Zero--dimensional Primary Decomposition 4.3 Higher Dimensional Primary Decomposition 4.4 The Equidimensional Part of an Ideal 4.5 The Radical 4.6 Procedures5. Hilbert Function 5.1 The Hilbert Function and the Hilbert Polynomial 5.2 Examples and Computation of the Hilbert--Poincare Series 5.3 Properties of the Hilbert Polynomial 5.4 Filtrations and the Lemma of Artin--Rees 5.5 The Hilbert--Samuel Function 5.6 Characterization of the Dimension of Local Rings 5.7 Singular Locus6. Complete Local Rings 6.1 Formal Power Series Rings 6.2 Weierstrass Preparation Theorem 6.3 Completions 6.4 Standard bases7. Homological Algebra 7.1 Tor 7.2 Fitting Ideals 7.3 Flatness 7.4 Local Criteria for Flatness 7.5 Flatness and Standard Bases 7.6 Koszul Complex 7.7 Cohen-Macaulay Rings 7.8 Further Characterization of Cohen-MacaulaynessA. Geometric Background A.1 Introduction by Pictures A.2 Affine algebraic varieties A.3 Spectrum and Affine Schemes A.4 Projective Varieties and Projective Schemes A.5 Morphisms between Varieties A.6 Projective Morphisms and elimination A.7 Local versus Global Properties A.8 SingularitiesB. SINGULAR - A Short Introduction B.1 Downloading Instructions B.2 Getting Started B.3 Procedures and Libraries B.4 Data Types B.5 Functions B.6 Control Structures B.7 System variablesSINGULAR Reference Manual Index Glossary
SynopsisThis book can be understood as a model for teaching commutative algebra, taking into account modern developments such as algorithmic and computational aspects. As soon as a new concept is introduced, it is shown how to handle it by computer. The computations are exemplified with the computer algebra system Singulardeveloped by the authors. Singularis a special system for polynomial computation with many features for global as well as for local commutative algebra and algebraic geometry. The book includes a CD with a version of Singularfor various platforms (Unix/Linux, Windows, Macintosh), including all examples and procedures explained in the book. The book can be used for courses, seminars and as a basis for studying research papers in commutative algebra, computer algebra and algebraic geometry., This book can be understood as a model for teaching commutative algebra, and takes into account modern developments such as algorithmic and computational aspects. As soon as a new concept is introduced, the authors show how the concept can be worked on using a computer. The computations are exemplified with the computer algebra system Singular, developed by the authors. Singular is a special system for polynomial computation with many features for global as well as for local commutative algebra and algebraic geometry. The book includes a CD containing Singular as well as the examples and procedures explained in the book.
LC Classification NumberQA251.3.G745 2002
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