Solving Polynomial Equations: Foundations, Algorithms, and Applications

Front Cover
Alicia Dickenstein, Ioannis Z. Emiris
Springer Science & Business Media, Apr 27, 2005 - Computers - 424 pages
0 Reviews
The subject of this book is the solution of polynomial equations, that is, s- tems of (generally) non-linear algebraic equations. This study is at the heart of several areas of mathematics and its applications. It has provided the - tivation for advances in di?erent branches of mathematics such as algebra, geometry, topology, and numerical analysis. In recent years, an explosive - velopment of algorithms and software has made it possible to solve many problems which had been intractable up to then and greatly expanded the areas of applications to include robotics, machine vision, signal processing, structural molecular biology, computer-aided design and geometric modelling, as well as certain areas of statistics, optimization and game theory, and b- logical networks. At the same time, symbolic computation has proved to be an invaluable tool for experimentation and conjecture in pure mathematics. As a consequence, the interest in e?ective algebraic geometry and computer algebrahasextendedwellbeyonditsoriginalconstituencyofpureandapplied mathematicians and computer scientists, to encompass many other scientists and engineers. While the core of the subject remains algebraic geometry, it also calls upon many other aspects of mathematics and theoretical computer science, ranging from numerical methods, di?erential equations and number theory to discrete geometry, combinatorics and complexity theory. Thegoalofthisbookistoprovideageneralintroduction tomodernma- ematical aspects in computing with multivariate polynomials and in solving algebraic systems.
  

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Introduction to residues and resultants
1
111 Local analytic residue
3
12 Some applications of residues
8
131 Definition
16
141 Systems of equations in two variables
21
15 Multidimensional residues
27
16 Multivariate resultants
44
17 Residues and resultants
55
Tools for computing primary decompositions
219
Putting it all together
228
Algorithms and their complexities
241
61 Statement of the problems
242
62 Algorithms and complexity
247
63 Dense encoding and algorithms
248
641 Basic definitions and examples
255
65 The NewtonHensel method
263

Solving equations via algebras
63
21 Solving equations
64
22 Ideals defined by linear conditions
78
23 Resultants
91
24 Factoring
100
25 Galois theory
114
Symbolicnumeric methods for solving polynomial equations and applications
125
31 Solving polynomial systems
126
32 Structure of the quotient algebra
131
33 Duality
141
34 Resultant constructions
145
35 Geometric solvers
151
36 Applications
158
An algebraists view on border bases
169
41 Commuting endomorphisms
172
42 Border prebases
179
43 Border bases
186
44 Application to statistics
195
Tools for computing primary decompositions and applications to ideals associated to Bayesian networks
203
Algebraic varieties and components
205
Bayesian networks and Markov ideals
212
66 Other trends
266
Toric resultants and applications to geometric modelling
269
71 Toric elimination theory
270
72 Matrix formulae
279
73 Implicitization with base points
288
74 Implicit support
292
75 Algebraic solving by linear algebra
298
Introduction to numerical algebraic geometry
301
80 Introduction
302
81 Homotopy continuation methods an overview
303
82 Homotopies to approximate all isolated solutions
305
83 Homotopies for positive dimensional solution sets
326
84 Software and applications
335
Four lectures on polynomial absolute factorization
339
Theorems of Hilbert and Bertini reduction to the bivariate case irreducibility tests 911 Hilberts irreducibility
344
Factorization algorithms via computations in algebraic number fields
351
Factorization algorithms via computations in the complex plane
358
Reconstruction of the exact factors
378
References
393
Index
419
Copyright

Common terms and phrases

References to this book

All Book Search results »

Bibliographic information