Solving Polynomial Equations: Foundations, Algorithms, and Applications (Algorithms and Computation in Mathematics (14))

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 modelling, 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. 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.

this Book Provides A General Introduction To Modern Mathematical Aspects In Computing With Multivariate Polynomials And In Solving Algebraic Systems. It Presents The State Of The Art In Several Symbolic, Numeric, And Symbolic-numeric Techniques, Including Effective And Algorithmic Methods In Algebraic Geometry And Computational Algebra, Complexity Issues, And Applications Ranging From Statistics And Geometric Modelling To Robotics And Vision.

graduate Students, As Well As Researchers In Related Areas, Will Find An Excellent Introduction To Currently Interesting Topics. These Cover Groebner And Border Bases, Multivariate Resultants, Residues, Primary Decomposition, Multivariate Polynomial Factorization, Homotopy Continuation, Complexity Issues, And Their Applications.

"This book provides a general introduction to modern mathematical aspects in computing with multivariate polynomials and in solving algebraic systems. It presents the state-of-the-art in several symbolic, numeric, and symbolic-numeric techniques, including effective and algorithmic methods in algebraic geometry and computational algebra, complexity issues, and applications ranging from statistics and geometric modelling to robotics and vision." "Graduate students, as well as researchers in related areas, will find an excellent introduction to currently interesting topics. These cover Groebner and border bases, multivariate resultants residues, primary decomposition, multivariate polynomial factorization, homotopy continuation, complexity issues, and their applications."--Jacket Introduction to residues and resultants....Pages 1-61 Solving equations via algebras....Pages 63-123 Symbolic-numeric methods for solving polynomial equations and applications....Pages 125-168 An algebraist’s view on border bases....Pages 169-202 Tools for computing primary decompositions and applications to ideals associated to Bayesian networks....Pages 203-239 Algorithms and their complexities....Pages 241-268 Toric resultants and applications to geometric modelling....Pages 269-300 Introduction to numerical algebraic geometry....Pages 301-337 Four lectures on polynomial absolute factorization....Pages 339-392 "This book provides a general introduction to modern mathematical aspects in computing with multivariate polynomials and in solving algebraic systems. It presents the state-of-the-art in several symbolic, numeric, and symbolic-numeric techniques, including effective and algorithmic methods in algebraic geometry and computational algebra, complexity issues, and applications ranging from statistics and geometric modelling to robotics and vision."--Publisher description