Solving Polynomial Equations

1. Front Matter

   Pages I-XIII

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2. Introduction to residues and resultants

   • Eduardo Cattani, Alicia Dickenstein

   Pages 1-61

3. Solving equations via algebras

   • David A. Cox

   Pages 63-123

4. Symbolic-numeric methods for solving polynomial equations and applications

   • Mohamed Elkadi, Bernard Mourrain

   Pages 125-168

5. An algebraist’s view on border bases

   • Achim Kehrein, Martin Kreuzer, Lorenzo Robbiano

   Pages 169-202

6. Tools for computing primary decompositions and applications to ideals associated to Bayesian networks

   • Michael Stillman

   Pages 203-239

7. Algorithms and their complexities

   • Juan Sabia

   Pages 241-268

8. Toric resultants and applications to geometric modelling

   • Ioannis Z. Emiris

   Pages 269-300

9. Introduction to numerical algebraic geometry

   • Andrew J. Sommese, Jan Verschelde, Charles W. Wampler

   Pages 301-337

10. Four lectures on polynomial absolute factorization

    • Guillaume Chèze, André Galligo

    Pages 339-392

11. Back Matter

    Pages 393-425

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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.

• algorithm
• algorithms
• complexity
• computer algebra
• polynomial
• robotics
• statistics
• system solving

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)

• Departamento de Matemática Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina

  Alicia Dickenstein

• Department of Informatics and Telecommunications, National Kapodistrian University of Athens, Panepistimiopolis, Greece

  Ioannis Z. Emiris

Policies and ethics