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Algorithmic algebra

Author: Bhubaneswar Mishra
Publisher: New York : Springer-Verlag, ©1993.
Series: Texts and monographs in computer science.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"Algorithmic Algebra studies some of the main algorithmic tools of computer algebra, covering such topics as Gröbner bases, characteristic sets, resultants and semialgebraic sets. The main purpose of the book is to acquaint advanced undergraduate and graduate students in computer science, engineering and mathematics with the algorithmic ideas in computer algebra so that they could do research in computational  Read more...
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Bhubaneswar Mishra
ISBN: 0387940901 9780387940908 3540940901 9783540940906
OCLC Number: 28218479
Description: xii, 416 pages : illustrations ; 24 cm.
Contents: 1. Introduction --
2. Algebraic Preliminaries --
3. Computational Ideal Theory --
4. Solving Systems of Polynomial Equations --
5. Characteristic Sets --
6. An Algebraic Interlude --
7. Resultants and Subresultants --
8. Real Algebra. 1. Introduction --
1.1. Prologue: Algebra and Algorithms --
1.2. Motivations --
1.2.1. Constructive Algebra --
1.2.2. Algorithmic and Computational Algebra --
1.2.3. Symbolic Computation --
1.2.4. Applications --
1.3. Algorithmic Notations --
1.3.1. Data Structures --
1.3.2. Control Structures --
1.4. Epilogue --
2. Algebraic Preliminaries --
2.1. Introduction to Rings and Ideals --
2.1.1. Rings and Ideals --
2.1.2. Homomorphism, Contraction and Extension --
2.1.3. Ideal Operations --
2.2. Polynomial Rings --
2.2.1. Dickson's Lemma --
2.2.2. Admissible Orderings on Power Products --
2.3. Gröbner Bases --
2.3.1. Gröbner Bases in K[x1, x2, ..., xn] --
2.3.2. Hilbert's Basis Theorem --
2.3.3. Finite Gröbner Bases --
2.4. Modules and Syzygies --
2.5. S-Polynomials --
3. Computational Ideal Theory --
3.1. Introduction --
3.2. Strongly Computable Ring --
3.2.1. Example: Computable Field --
3.2.2. Example: Ring of Integers --
3.3. Head Reductions and Gröbner Bases --
3.3.1. Algorithm to Compute Head Reduction --
3.3.2. Algorithm to Compute Gröbner Bases --
3.4. Detachability Computation --
3.4.1. Expressing with the Gröbner Basis --
3.4.2. Detachability --
3.5. Syzygy Computation --
3.5.1. Syzygy of a Gröbner Basis: Special Case --
3.5.2. Syzygy of a Set: General Case --
3.6. Hilbert's Basis Theorem: Revisited --
3.7. Applications of Gröbner Bases Algorithms --
3.7.1. Membership --
3.7.2. Congruence, Subideal and Ideal Equality --
3.7.3. Sum and Product --
3.7.4. Intersection --
3.7.5. Quotient --
4. Solving Systems of Polynomial Equations --
4.1. Introduction --
4.2. Triangular Set --
4.3. Some Algebraic Geometry --
4.3.1. Dimension of an Ideal --
4.3.2. Solvability: Hilbert's Nullstellensatz --
4.3.3. Finite Solvability --
4.4. Finding the Zeros --
5. Characteristic Sets --
5.1. Introduction --
5.2. Pseudodivision and Successive Pseudodivision --
5.3. Characteristic Sets --
5.4. Properties of Characteristic Sets --
5.5. Wu-Ritt Process --
5.6. Computation --
5.7. Geometric Theorem Proving --
6. An Algebraic Interlude --
6.1. Introduction --
6.2. Unique Factorization Domain --
6.3. Principal Ideal Domain --
6.4. Euclidean Domain --
6.5. Gauss Lemma --
6.6. Strongly Computable Euclidean Domains --
7. Resultants and Subresultants --
7.1. Introduction --
7.2. Resultants --
7.3. Homomorphisms and Resultants --
7.3.1. Evaluation Homomorphism --
7.4. Repeated Factors in Polynomials and Discriminants --
7.5. Determinant Polynomial --
7.5.1. Pseudodivision: Revisited --
7.5.2. Homomorphism and Pseudoremainder --
7.6. Polynomial Remainder Sequences --
7.7. Subresultants --
7.7.1. Subresultants and Common Divisors --
7.8. Homomorphisms and Subresultants --
7.9. Subresultant Chain --
7.10. Subresultant Chain Theorem --
7.10.1. Habicht's Theorem --
7.10.2. Evaluation Homomorphisms --
7.10.3. Subresultant Chain Theorem --
8. Real Algebra --
8.1. Introduction --
8.2. Real Closed Fields --
8.3. Bounds on the Roots --
8.4. Sturm's Theorem --
8.5. Real Algebraic Numbers --
8.5.1. Real Algebraic Number Field --
8.5.2. Root Separation, Thorn's Lemma and Representation --
8.6. Real Geometry --
8.6.1. Real Algebraic Sets --
8.6.2. Delineability --
8.6.3. Tarski-Seidenberg Theorem --
8.6.4. Representation and Decomposition of Semialgebraic Sets --
8.6.5. Cylindrical Algebraic Decomposition --
8.6.6. Tarski Geometry.
Series Title: Texts and monographs in computer science.
Responsibility: Bhubaneswar Mishra.
More information:

Abstract:

Based on a graduate computer science course entitled "Symbolic Computational Algebra" taught by the author at New York University, this book covers four main topics - Grobner bases, characteristic  Read more...

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