Gr¶bner Bases

A Computational Approach to Commutative Algebra

By: H. Kredel, Thomas Becker, Volker Weispfenning

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Hardcover | 23 March 1998 | Edition Number 2

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604 Pages

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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.

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Preface
How to Use This Book
List of Algorithms
Basics	p. 1
Commutative Rings with Unity	p. 15
Why Abstract Algebra?	p. 15
Groups	p. 16
Rings	p. 19
Subrings and Homomorphisms	p. 22
Ideals and Residue Class Rings	p. 25
The Homomorphism Theorem	p. 32
Gcd's, Lcm's, and Principal Ideal Domains	p. 38
Maximal and Prime Ideals	p. 47
Prime Rings and Characteristic	p. 49
Adjunction, Products, and Quotient Rings	p. 51
Polynomial Rings	p. 61
Definitions	p. 61
Euclidean Domains	p. 78
Unique Factorization Domains	p. 87
The Gaussian Lemma	p. 92
Polynomial Gcd's	p. 97
Squarefree Decomposition of Polynomials	p. 99
Factorization of Polynomials	p. 109
The Chinese Remainder Theorem	p. 114
Vector Spaces and Modules	p. 119
Vector Spaces	p. 119
Independent Sets and Dimension	p. 129
Modules	p. 133
Orders and Abstract Reduction Relations	p. 141
The Axiom of Choice and Some Consequences in Algebra	p. 141
Relations	p. 149
Foundedness Properties	p. 155
Some Special Orders	p. 167
Reduction Relations	p. 173
Computing in Algebraic Structures	p. 178
Grobner Bases	p. 187
Term Orders and Polynomial Reductions	p. 188
Grobner Bases - Existence and Uniqueness	p. 204
Grobner Bases - Construction	p. 210
Standard Representations	p. 218
Improved Grobner Basis Algorithms	p. 222
The Extended Grobner Basis Algorithm	p. 237
First Applications of Grobner Bases	p. 243
Computation of Syzygies	p. 243
Basic Algorithms in Ideal Theory	p. 256
Dimension of Ideals	p. 270
Uniform Word Problems	p. 276
Field Extensions and the Hilbert Nullstellensatz	p. 293
Field Extensions	p. 293
The Algebraic Closure of a Field	p. 306
Separable Polynomials and Perfect Fields	p. 309
The Hilbert Nullstellensatz	p. 312
Height and Depth of Prime Ideals	p. 323
Implicitization of Rational Parametrizations	p. 327
Invertibility of Polynomial Maps	p. 330
Decomposition, Radical, and Zeroes of Ideals	p. 335
Preliminaries	p. 335
The Radical of a Zero-Dimensional Ideal	p. 340
The Number of Zeroes of an Ideal	p. 346
Primary Ideals	p. 350
Primary Decomposition in Noetherian Rings	p. 358
Primary Decomposition of Zero-Dimensional Ideals	p. 366
Radical and Decomposition in Higher Dimensions	p. 388
Computing Real Zeroes of Polynomial Systems	p. 399
Linear Algebra in Residue Class Rings	p. 423
Grobner Bases and Reduced Terms	p. 423
Computing in Finitely Generated Algebras	p. 433
Dimensions and the Hilbert Function	p. 441
Variations on Grobner Bases	p. 453
Grobner Bases over PID's and Euclidean Domains	p. 453
Homogeneous Grobner Bases	p. 466
Homogenization	p. 477
Grobner Bases for Polynomial Modules	p. 485
Systems of Linear Equations	p. 488
Standard Bases and the Tangent Cone	p. 496
Symmetric Functions	p. 501
Appendix: Outlook on Advanced and Related Topics	p. 511
Complexity of Grobner Basis Constructions	p. 511
Term Orders and Universal Grobner Bases	p. 514
Comprehensive Grobner Bases	p. 515
Grobner Bases and Automatic Theorem Proving	p. 518
Characteristic Sets and Wu-Ritt Reduction	p. 520
Term Rewriting	p. 523
Standard Bases in Power Series Rings	p. 525
Non-Commutative Grobner Bases	p. 526
Grobner Bases and Differential Algebra	p. 528
Selected Bibliography	p. 531
List of Symbols	p. 561
Index	p. 567
Table of Contents provided by Blackwell. All Rights Reserved.

Gr¶bner Bases

A Computational Approach to Commutative Algebra

At a Glance

Hardcover

Shipping

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