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Tables of Contents for Fundamental Problems of Algorithmic Algebra
Chapter/Section Title
Page #
Page Count
Preface
xiii
 
Current Textbooks in Algorithmic Algebra
xv
 
Introduction
1
484
Fundamental Problem of Algebra
2
2
Fundamental Problem of Classical Algebraic Geometry
4
2
Fundamental Problem of Ideal Theory
6
3
Representation and Size
9
1
Computational Models
10
3
Asymptotic Notations
13
2
Complexity of Multiplication
15
3
On Bit Versus Algebraic Complexity
18
2
Miscellany
20
6
Computer Algebra Systems
26
1
Arithmetic
27
16
The Discrete Fourier Transform
28
4
Polynomial Multiplication
32
2
Modular Fast Fourier Transform
34
3
Fast Integer Multiplication
37
4
Matrix Multiplication
41
2
The Greatest Common Denominator
43
34
Unique Factorization Domain
44
3
Euclid's Algorithm
47
3
Euclidean Ring
50
4
The Half-Greatest Common Denominator Problem
54
3
Properties of the Norm
57
4
Polynomial Half-GCD
61
16
Integer Half-GCD
67
10
Subresultants
77
27
Primitive Factorization
78
4
Pseudoremainders and Polynomial Remainder Sequence
82
2
Determinantal Polynomials
84
3
Polynomial Pseudoquotient
87
2
The Subresultant Polynomial Remainder Sequence
89
1
Subresultants
90
2
Pseudosubresultants
92
5
Subresultant Theorem
97
4
Correctness of the Subresultant Polynomial Remainder Sequence Algorithm
101
3
Modular Techniques
104
20
Chinese Remainder Theorem
104
3
Evaluation and Interpolation
107
4
Finding Prime Moduli
111
2
Lucky Homomorphisms for the GCD
113
3
Coefficient Bounds for Factors
116
4
A Modular Greatest Common Denominator Algorithm
120
2
What Else in GCD Computation?
122
2
Fundamental Theorem of Algebra
124
17
Elements of Field Theory
125
4
Ordered Rings
129
1
Formally Real Rings
130
2
Constructible Extensions
132
3
Real Closed Fields
135
3
Fundamental Theorem of Algebra
138
3
Roots of Polynomials
141
45
Elementary Properties of Polynomial Roots
142
5
Root Bounds
147
4
Algebraic Numbers
151
4
Resultants
155
6
Symmetric Functions
161
6
Discriminant
167
2
Root Separation
169
4
A Generalized Hadamard Bound
173
5
Isolating Intervals
178
2
On Newton's Method
180
2
Guaranteed Convergence of Newton Iteration
182
4
Sturm Theory
186
33
Sturm Sequences From Polynomial Remainder Sequences
187
3
A Generalized Sturm Theorem
190
5
Corollaries and Applications
195
6
Integer and Complex Roots
201
3
The Routh-Hurwitz Theorem
204
5
Sign Encoding of Algebraic Numbers: Thom's Lemma
209
2
Problem of Relative Sign Conditions
211
3
The BKR Algorithm
214
5
Gaussian Lattice Reduction
219
15
Lattices
219
5
Shortest Vectors in Planar Lattices
224
3
Coherent Remainder Sequences
227
7
Lattice Reduction and Applications
234
24
Gram-Schmidt Orthogonalization
235
4
Minkowski's Convex Body Theorem
239
3
Weakly Reduced Bases
242
1
Reduced Bases and the LLL Algorithm
243
4
Short Vectors
247
4
Factorization Via Reconstruction of Minimal Polynomials
251
7
Linear Systems
258
42
Sylvester's Identity
259
3
Fraction-Free Determinant Computation
262
7
Matrix Inversion
269
2
Hermite Normal Form
271
4
A Multiple GCD Bound and Algorithm
275
5
Hermite Reduction Step
280
6
Bachem-Kannan Algorithm
286
6
Smith Normal Form
292
4
Further Applications
296
4
Elimination Theory
300
63
Hilbert Basis Theorem
301
3
Hilbert Nullstellensatz
304
4
Specializations
308
5
Resultant Systems
313
5
Sylvester Resultant Revisited
318
3
Inertial Ideal
321
6
The Macaulay Resultant
327
7
U-Resultant
334
3
Generalized Characteristic Polynomial
337
5
Generalized U-Resultant
342
8
A Multivariate Root Bound
350
13
Power Series
355
4
Counting Irreducible Polynomias
359
4
Grobner Bases
363
35
Admissible Orderings
364
8
Normal Form Algorithm
372
6
Characterizations of Grobner Bases
378
4
Buchberger's Algorithm
382
2
Uniqueness
384
2
Elimination Properties
386
5
Computing in Quotient Rings
391
2
Syzygies
393
5
Bounds in Polynomial Ideal Theory
398
48
Some Bounds in Polynomial Ideal Theory
399
2
The Hilbert-Serre Theorem
401
6
Homogeneous Sets
407
5
Cone Decomposition
412
4
Exact Decomposition of NF(I)
416
7
Exact Decomposition of Ideals
423
1
Bounding the Macaulay Constants
424
4
Term-Rewriting Systems
428
4
A Quadratic Counter
432
4
Uniqueness Property
436
2
Lower Bounds
438
8
Properties of S0
442
4
Continued Fractions
446
39
Introduction
447
2
Extended Numbers
449
2
General Terminology
451
4
Ordinary Continued Fractions
455
5
Continued Fractions as Mobius Transformations
460
5
Convergence Properties
465
5
Real Mobius Transformations
470
4
Continued Fractions of Roots
474
4
Arithmetic Operations
478
7
References
485
10
Index
495
13
Index to Symbols
508