Dickenstein A., Emiris I.Z. (eds.) Solving Polynomial Equations. Foundations, Algorithms, and Applications

Springer, 2005, -432 p.The subject of this book is the solution of polynomial equations, that is, systems of (generally) non-linear algebraic equations. This study is at the heart of several areas of mathematics and its applications. It has provided the motivation for advances in different branches of mathematics such as algebra, geometry, topology, and numerical analysis. In recent years, an explosive development 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 modeling, as well as certain areas of statistics, optimization and game theory, and biological 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 effective algebraic geometry and computer algebra has extended well beyond its original constituency of pure and applied 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, differential equations and number theory to discrete geometry, combinatorics and complexity theory.
The goal of this book is to provide a general introduction to modern mathematical aspects in computing with multivariate polynomials and in solving algebraic systems. It is aimed to upper-level undergraduate and graduate students, and researchers in pure and applied mathematics and engineering, interested in computational algebra and in the connections between computer algebra and numerical mathematics. Most chapters assume a solid grounding in linear algebra while for several of them a basic knowledge of Gr¨obner bases, at the level of [CLO97] is expected. Gr¨obner bases have become a basic standard tool in computer algebra and the reader may consult any other textbook such as [AL94, BW93, CLO98, GP02], or the introductory chapter in [CCS99]. Below we discuss briefly the content of each chapter and some of their prerequisites.
The book describes foundations, recent developments and applications of Grobner and border bases, residues, multivariate resultants, including toric elimination theory, primary decomposition of ideals, multivariate polynomial factorization, as well as homotopy continuation methods. While some of the chapters are introductory in nature, others present the state-of-the-art in symbolic techniques in polynomial system solving, including effective and algorithmic methods in algebraic geometry and computational algebra, complexity issues, and applications.We also discuss several numeric and symbolic-numeric methods. This is not a standard textbook in that each chapter is independent and, largely, self-contained. However, there are strong links between the different chapters as evidenced by the many cross-references. While the reader gains the advantage of being able to access the book at many different places and of seeing the interplay of different views of the same concepts, we should note that, because of the different needs and traditions, some notations inevitably vary between different chapters. We have tried to note this in the text whenever it occurs. The single bibliography and index underline the unity of the subject.Introduction to residues and resultants
Solving equations via algebras
Symbolic-numeric methods for solving polynomial equations and applications
An algebraist’s view on border bases
Tools for computing primary decompositions and applications to ideals associated to Bayesian networks
Algorithms and their complexities
Toric resultants and applications to geometric modeling
Introduction to numerical algebraic geometry
Four lectures on polynomial absolute factorization