Thomas Becker, Volker Weispfenning

Gröbner Bases (eBook, PDF)

A Computational Approach to Commutative Algebra

• Format: PDF

Produktbeschreibung

This book provides a comprehensive treatment of Gr bner bases theory embedded in an introduction to commutative algebra from a computational point of view. The book explains how the Buchberger algorithm and the theory surrounding it are eminently important both for the mathematical theory and for computational applications. A number of results such as optimized version of the Buchberger algorithm are presented in textbook format for the first time.

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Produktdetails

• Verlag: Springer US
• Seitenzahl: 576
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9781461209133
• Artikelnr.: 44052293

Inhaltsangabe

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 RationalParametrizations.
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.

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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 RationalParametrizations.- 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.