Written by the founders of the new and expanding field of numerical algebraic geometry, this is the first book that uses an algebraic-geometric approach to the numerical solution of polynomial systems and also the first one to treat numerical methods for finding positive dimensional solution sets. The text covers the full theory from methods developed for isolated solutions in the 1980’s to the most recent research on positive dimensional sets.

Contents:

• Background:
• Polynomial Systems
• Homotopy Continuation
• Projective Spaces
• Genericity and Probability One
• Polynomials of One Variable
• Other Methods
• Isolated Solutions:
• Coefficient-Parameter Homotopy
• Polynomial Structures
• Case Studies
• Endpoint Estimation
• Checking Results and Other Implementation Tips
• Positive Dimensional Solutions:
• Basic Algebraic Geometry
• Basic Numerical Algebraic Geometry
• A Cascade Algorithm for Witness Supersets
• The Numerical Irreducible Decomposition
• The Intersection of Algebraic Sets
• Appendices:
• Algebraic Geometry
• Software for Polynomial Continuation
• HomLab User’s Guide

“The text is written in a very smooth and intelligent form, yielding a readable book whose contents are accessible to a wide class of readers, even to undergraduate students, provided that they accept that some delicate points of some of the proofs could be omitted. Its readability and fast access to the core of the book makes it recommendable as a pleasant read.”

“This is an excellent book on numerical solutions of polynomials systems for engineers, scientists and numerical analysts. As pioneers of the field of numerical algebraic geometry, the authors have provided a comprehensive summary of ideas, methods, problems of numerical algebraic geometry and applications to solving polynomial systems. Through the book readers will experience the authors' original ideas, contributions and their techniques in handling practical problems ... Many interesting examples from engineering and science have been used throughout the book. Also the exercises are well designed in line with the content, along with the algorithms, sample programs in Matlab and author's own software 'HOMLAB' for polynomial continuation. This is a remarkable book that I recommend to engineers, scientists, researchers, professionals and students, and particularly numerical analysts who will benefit from the rapid development of numerical algebraic geometry.”