# Solving Polynomial Equations: Foundations, Algorithms, and Applications

Springer Science & Business Media, Apr 27, 2005 - Computers - 424 pages
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.

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### 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