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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.
List of contents
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 modelling.- to numerical algebraic geometry.- Four lectures on polynomial absolute factorization.
Summary
This useful collection 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 a range of applications. Graduate students, as well as researchers in related areas, will find an excellent introduction to currently interesting topics.
Additional text
From the reviews of the first edition:
"[...] The editors of the book — Alicia Dickenstein and Ioannis Emiris — have done a very good job in collecting a set of expository lectures on a very interesting branch of mathematics. I imagine that this book will be of use to anyone working in the area, and would be a good introduction for a graduate student or someone wishing to start working in the field. [...]"
Darren Glass, MAA Review, January 2006
"Dieses Buch besteht aus neun weitgehend unabhängigen Kapiteln zum Thema "Lösung polynomialer Gleichungssysteme", geschrieben von bekannten Experten. Der Schwerpunkt liegt auf spezialisierten Themen, bei denen in jüngerer Zeit beachtliche Fortschritte erzielt wurden. Es gibt deshalb wenig Überschneidungen mit anderen Büchern der Computeralgebra. [...] Insgesamt ist dieses Buch eine reiche Quelle von Informationen und eine sehr willkommene Ergänzung zu den bereits vorhandenen Lehrbüchern im Bereich der Lösung polynomialer Gleichungssysteme. Es sollte in keinem Regal fehlen."
Peter Bürgisser, Computeralgebra, Ausgabe 39, Oktober 2006
"Although about 15 authors have contributed to this book it constitutes a unified whole. Its subjects are the diverse methods, techniques and algorithms in solving multivariate (non-linear) polynomial equations or systems of them, which mostly have been developed in recent years. … All in all there is presented a detailed account, which often leads to the front of research." (G. Kowol, Monatshefte für Mathematik, Vol. 148 (4), 2006)
Report
From the reviews of the first edition:
"[...] The editors of the book - Alicia Dickenstein and Ioannis Emiris - have done a very good job in collecting a set of expository lectures on a very interesting branch of mathematics. I imagine that this book will be of use to anyone working in the area, and would be a good introduction for a graduate student or someone wishing to start working in the field. [...]"
Darren Glass, MAA Review, January 2006
"Dieses Buch besteht aus neun weitgehend unabhängigen Kapiteln zum Thema "Lösung polynomialer Gleichungssysteme", geschrieben von bekannten Experten. Der Schwerpunkt liegt auf spezialisierten Themen, bei denen in jüngerer Zeit beachtliche Fortschritte erzielt wurden. Es gibt deshalb wenig Überschneidungen mit anderen Büchern der Computeralgebra. [...] Insgesamt ist dieses Buch eine reiche Quelle von Informationen und eine sehr willkommene Ergänzung zu den bereits vorhandenen Lehrbüchern im Bereich der Lösung polynomialer Gleichungssysteme. Es sollte in keinem Regal fehlen."
Peter Bürgisser, Computeralgebra, Ausgabe 39, Oktober 2006
"Although about 15 authors have contributed to this book it constitutes a unified whole. Its subjects are the diverse methods, techniques and algorithms in solving multivariate (non-linear) polynomial equations or systems of them, which mostly have been developed in recent years. ... All in all there is presented a detailed account, which often leads to the front of research." (G. Kowol, Monatshefte für Mathematik, Vol. 148 (4), 2006)
