Shortcuts
Please wait while page loads.
SISSA Library . Default .
PageMenu- Main Menu-
Page content

Catalogue Display

Gröbner Bases: A Computational Approach to Commutative Algebra /

Gröbner Bases: A Computational Approach to Commutative Algebra /
Catalogue Information
Field name Details
Dewey Class 512.2
Title Gröbner Bases ([EBook] :) : A Computational Approach to Commutative Algebra / / by Thomas Becker, Volker Weispfenning.
Author Becker, Thomas
Added Personal Name Weispfenning, Volker author.
Other name(s) SpringerLink (Online service)
Publication New York, NY : : Springer New York : : Imprint: Springer, , 1993.
Physical Details XXII, 576 p. : online resource.
Series Graduate texts in mathematics 0072-5285 ; ; 141
ISBN 9781461209133
Summary Note 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.:
Contents note 0 Basics -- 0.1 Natural Numbers and Integers -- 0.2 Maps -- 0.3 Mathematical Algorithms -- Notes -- 1 Commutative Rings with Unity -- 1.1 Why Abstract Algebra? -- 1.2 Groups -- 1.3 Rings -- 1.4 Subrings and Homomorphisms -- 1.5 Ideals and Residue Class Rings -- 1.6 The Homomorphism Theorem -- 1.7 Gcd’s, Lcm’s, and Principal Ideal Domains -- 1.8 Maximal and Prime Ideals -- 1.9 Prime Rings and Characteristic -- 1.10 Adjunction, Products, and Quotient Rings -- Notes -- 2 Polynomial Rings -- 2.1 Definitions -- 2.2 Euclidean Domains -- 2.3 Unique Factorization Domains -- 2.4 The Gaussian Lemma -- 2.5 Polynomial Gcd’s -- 2.6 Squarefree Decomposition of Polynomials -- 2.7 Factorization of Polynomials -- 2.8 The Chinese Remainder Theorem -- Notes -- 3 Vector Spaces and Modules -- 3.1 Vector Spaces -- 3.2 Independent Sets and Dimension -- 3.3 Modules -- Notes -- 4 Orders and Abstract Reduction Relations -- 4.1 The Axiom of Choice and Some Consequences in Algebra -- 4.2 Relations -- 4.3 Foundedness Properties -- 4.4 Some Special Orders -- 4.5 Reduction Relations -- 4.6 Computing in Algebraic Structures -- Notes -- 5 Gröbner Bases -- 5.1 Term Orders and Polynomial Reductions -- 5.2 Gröbner Bases—Existence and Uniqueness -- 5.3 Gröbner Bases—Construction -- 5.4 Standard Representations -- 5.5 Improved Gröbner Basis Algorithms -- 5.6 The Extended Gröbner Basis Algorithm -- Notes -- 6 First Applications of Gröbner Bases -- 6.1 Computation of Syzygies -- 6.2 Basic Algorithms in Ideal Theory -- 6.3 Dimension of Ideals -- 6.4 Uniform Word Problems -- Notes -- 7 Field Extensions and the Hilbert Nullstellensatz -- 7.1 Field Extensions -- 7.2 The Algebraic Closure of a Field -- 7.3 Separable Polynomials and Perfect Fields -- 7.4 The Hilbert Nullstellensatz -- 7.5 Height and Depth of Prime Ideals -- 7.6 Implicitization of Rational Parametrizations -- 7.7 Invertibility of Polynomial Maps -- Notes -- 8 Decomposition, Radical, and Zeroes of Ideals -- 8.1 Preliminaries -- 8.2 The Radical of a Zero-Dimensional Ideal -- 8.3 The Number of Zeroes of an Ideal -- 8.4 Primary Ideals -- 8.5 Primary Decomposition in Noetherian Rings -- 8.6 Primary Decomposition of Zero-Dimensional Ideals -- 8.7 Radical and Decomposition in Higher Dimensions -- 8.8 Computing Real Zeroes of Polynomial Systems -- Notes -- 9 Linear Algebra in Residue Class Rings -- 9.1 Gröbner Bases and Reduced Terms -- 9.2 Computing in Finitely Generated Algebras -- 9.3 Dimensions and the Hilbert Function -- Notes -- 10 Variations on Gröbner Bases -- 10.1 Gröbner Bases over PID’s and Euclidean Domains -- 10.2 Homogeneous Gröbner Bases -- 10.3 Homogenization -- 10.4 Gröbner Bases for Polynomial Modules -- 10.5 Systems of Linear Equations -- 10.6 Standard Bases and the Tangent Cone -- 10.7 Symmetric Functions -- Notes -- Appendix: Outlook on Advanced and Related Topics -- Complexity of Gröbner Basis Constructions -- Term Orders and Universal Gröbner Bases -- Comprehensive Gröbner Bases -- Gröbner Bases and Automatic Theorem Proving -- Characteristic Sets and Wu-Ritt Reduction -- Term Rewriting -- Standard Bases in Power Series Rings -- Non-Commutative Gröbner Bases -- Gröbner Bases and Differential Algebra -- Selected Bibliography -- Conference Proceedings -- Books and Monographs -- Articles -- List of Symbols.
System details note Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site http://dx.doi.org/10.1007/978-1-4612-0913-3
Links to Related Works
Subject References:
Authors:
Corporate Authors:
Series:
Classification:
Catalogue Information 42672 Beginning of record . Catalogue Information 42672 Top of page .

Reviews


This item has not been rated.    Add a Review and/or Rating42672
. E-mail This Page
Quick Search