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Tables of Contents for Ideals, Varieties, and Algorithms
Chapter/Section Title
Page #
Page Count
Preface to the First Edition
vii
Preface to the Second Edition
ix
Geometry, Algebra, and Algorithms
1
46
Polynomials and Affine Space
1
4
Affine Varieties
5
9
Parametrizations of Affine Varieties
14
15
Ideals
29
8
Polynomials of One Variable
37
10
Groebner Bases
47
65
Introduction
47
5
Orderings on the Monomials in k[x1,...,xn]
52
7
A Division Algorithm in k[x1,...,xn]
59
8
Monomial Ideals and Dickson's Lemma
67
6
The Hilbert Basis Theorem and Groebner Bases
73
6
Properties of Groebner Bases
79
7
Buchberger's Algorithm
86
7
First Applications of Groebner Bases
93
6
(Optional) Improvements on Buchberger's Algorithm
99
13
Elimination Theory
112
55
The Elimination and Extension Theorems
112
8
The Geometry of Elimination
120
4
Implicitization
124
9
Singular Points and Envelopes
133
13
Unique Factorization and Resultants
146
12
Resultants and the Extension Theorem
158
9
The Algebra-Geometry Dictionary
167
45
Hilbert's Nullstellensatz
167
6
Radical Ideals and the Ideal-Variety Correspondence
173
7
Sums, Products, and Intersections of Ideals
180
10
Zariski Closure and Quotients of Ideals
190
5
Irreducible Varieties and Prime Ideals
195
5
Decomposition of a Variety into Irreducibles
200
6
(Optional) Primary Decomposition of Ideals
206
4
Summary
210
2
Polynomial and Rational Functions on a Variety
212
49
Polynomial Mappings
212
6
Quotients of Polynomial Rings
218
8
Algorithmic Computations in k[x1,...,xn]/I
226
9
The Coordinate Ring of an Affine Variety
235
10
Rational Functions on a Variety
245
9
(Optional) Proof of the Closure Theorem
254
7
Robotics and Automatic Geometric Theorem Proving
261
50
Geometric Description of Robots
261
6
The Forward Kinematic Problem
267
7
The Inverse Kinematic Problem and Motion Planning
274
12
Automatic Geometric Theorem Proving
286
16
Wu's Method
302
9
Invariant Theory of Finite Groups
311
38
Symmetric Polynomials
311
10
Finite Matrix Groups and Rings of Invariants
321
8
Generators for the Ring of Invariants
329
9
Relations Among Generators and the Geometry of Orbits
338
11
Projective Algebraic Geometry
349
80
The Projective Plane
349
11
Projective Space and Projective Varieties
360
10
The Projective Algebra-Geometry Dictionary
370
8
The Projective Closure of an Affine Variety
378
6
Projective Elimination Theory
384
15
The Geometry of Quadric Hypersurfaces
399
13
Bezout's Theorem
412
17
The Dimension of a Variety
429
68
The Variety of a Monomial Ideal
429
4
The Complement of a Monomial Ideal
433
13
The Hilbert Function and the Dimension of a Variety
446
11
Elementary Properties of Dimension
457
8
Dimension and Algebraic Independence
465
8
Dimension and Nonsingularity
473
10
The Tangent Cone
483
14
Appendix A. Some Concepts from Algebra
497
4
§1. Fields and Rings
497
1
§2. Groups
498
1
§3. Determinants
499
2
Appendix B. Pseudocode
501
4
§1. Inputs, Outputs, Variables, and Constants
501
1
§2. Assignment Statements
502
1
§3. Looping Structures
502
1
§4. Branching Structures
503
2
Appendix C. Computer Algebra Systems
505
13
§1. AXIOM
505
3
§2. Maple
508
2
§3. Mathematica
510
2
§4. REDUCE
512
4
§5. Other Systems
516
2
Appendix D. Independent Projects
518
5
§1. General Comments
518
1
§2. Suggested Projects
518
5
References
523
4
Index
527