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Groebner Bases : A Computational Approach to Commutative Algebra - Thomas Becker

Groebner Bases

A Computational Approach to Commutative Algebra


Published: 23rd March 1998
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The origins of the mathematics in this book date back more than two thou­ sand years, as can be seen from the fact that one of the most important algorithms presented here bears the name of the Greek mathematician Eu­ clid. The word "algorithm" as well as the key word "algebra" in the title of this book come from the name and the work of the ninth-century scientist Mohammed ibn Musa al-Khowarizmi, who was born in what is now Uzbek­ istan and worked in Baghdad at the court of Harun al-Rashid's son. The word "algorithm" is actually a westernization of al-Khowarizmi's name, while "algebra" derives from "al-jabr," a term that appears in the title of his book Kitab al-jabr wa'l muqabala, where he discusses symbolic methods for the solution of equations. This close connection between algebra and al­ gorithms lasted roughly up to the beginning of this century; until then, the primary goal of algebra was the design of constructive methods for solving equations by means of symbolic transformations. During the second half of the nineteenth century, a new line of thought began to enter algebra from the realm of geometry, where it had been successful since Euclid's time, namely, the axiomatic method.

How to Use This Book
List of Algorithms
Basicsp. 1
Commutative Rings with Unityp. 15
Why Abstract Algebra?p. 15
Groupsp. 16
Ringsp. 19
Subrings and Homomorphismsp. 22
Ideals and Residue Class Ringsp. 25
The Homomorphism Theoremp. 32
Gcd's, Lcm's, and Principal Ideal Domainsp. 38
Maximal and Prime Idealsp. 47
Prime Rings and Characteristicp. 49
Adjunction, Products, and Quotient Ringsp. 51
Polynomial Ringsp. 61
Definitionsp. 61
Euclidean Domainsp. 78
Unique Factorization Domainsp. 87
The Gaussian Lemmap. 92
Polynomial Gcd'sp. 97
Squarefree Decomposition of Polynomialsp. 99
Factorization of Polynomialsp. 109
The Chinese Remainder Theoremp. 114
Vector Spaces and Modulesp. 119
Vector Spacesp. 119
Independent Sets and Dimensionp. 129
Modulesp. 133
Orders and Abstract Reduction Relationsp. 141
The Axiom of Choice and Some Consequences in Algebrap. 141
Relationsp. 149
Foundedness Propertiesp. 155
Some Special Ordersp. 167
Reduction Relationsp. 173
Computing in Algebraic Structuresp. 178
Grobner Basesp. 187
Term Orders and Polynomial Reductionsp. 188
Grobner Bases - Existence and Uniquenessp. 204
Grobner Bases - Constructionp. 210
Standard Representationsp. 218
Improved Grobner Basis Algorithmsp. 222
The Extended Grobner Basis Algorithmp. 237
First Applications of Grobner Basesp. 243
Computation of Syzygiesp. 243
Basic Algorithms in Ideal Theoryp. 256
Dimension of Idealsp. 270
Uniform Word Problemsp. 276
Field Extensions and the Hilbert Nullstellensatzp. 293
Field Extensionsp. 293
The Algebraic Closure of a Fieldp. 306
Separable Polynomials and Perfect Fieldsp. 309
The Hilbert Nullstellensatzp. 312
Height and Depth of Prime Idealsp. 323
Implicitization of Rational Parametrizationsp. 327
Invertibility of Polynomial Mapsp. 330
Decomposition, Radical, and Zeroes of Idealsp. 335
Preliminariesp. 335
The Radical of a Zero-Dimensional Idealp. 340
The Number of Zeroes of an Idealp. 346
Primary Idealsp. 350
Primary Decomposition in Noetherian Ringsp. 358
Primary Decomposition of Zero-Dimensional Idealsp. 366
Radical and Decomposition in Higher Dimensionsp. 388
Computing Real Zeroes of Polynomial Systemsp. 399
Linear Algebra in Residue Class Ringsp. 423
Grobner Bases and Reduced Termsp. 423
Computing in Finitely Generated Algebrasp. 433
Dimensions and the Hilbert Functionp. 441
Variations on Grobner Basesp. 453
Grobner Bases over PID's and Euclidean Domainsp. 453
Homogeneous Grobner Basesp. 466
Homogenizationp. 477
Grobner Bases for Polynomial Modulesp. 485
Systems of Linear Equationsp. 488
Standard Bases and the Tangent Conep. 496
Symmetric Functionsp. 501
Appendix: Outlook on Advanced and Related Topicsp. 511
Complexity of Grobner Basis Constructionsp. 511
Term Orders and Universal Grobner Basesp. 514
Comprehensive Grobner Basesp. 515
Grobner Bases and Automatic Theorem Provingp. 518
Characteristic Sets and Wu-Ritt Reductionp. 520
Term Rewritingp. 523
Standard Bases in Power Series Ringsp. 525
Non-Commutative Grobner Basesp. 526
Grobner Bases and Differential Algebrap. 528
Selected Bibliographyp. 531
List of Symbolsp. 561
Indexp. 567
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387979717
ISBN-10: 0387979719
Series: Graduate Texts in Mathematics
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 576
Published: 23rd March 1998
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 3.18
Weight (kg): 2.25
Edition Number: 2