Algorithms in Real Algebraic Geometry
Authors: Basu, Saugata, Pollack, Richard, Roy, MarieFrançoise
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 About this Textbook

The algorithmic problems of real algebraic geometry such as real root counting, deciding the existence of solutions of systems of polynomial equations and inequalities, finding global maxima or deciding whether two points belong in the same connected component of a semialgebraic set appear frequently in many areas of science and engineering. In this firstever graduate textbook on the algorithmic aspects of real algebraic geometry, the main ideas and techniques presented form a coherent and rich body of knowledge, linked to many areas of mathematics and computing.
Mathematicians already aware of real algebraic geometry will find relevant information about the algorithmic aspects, and researchers in computer science and engineering will find the required mathematical background.
Being selfcontained the book is accessible to graduate students and even, for invaluable parts of it, to undergraduate students.
This revised second edition contains several recent results, notably on discriminants of symmetric matrices, real root isolation, global optimization, quantitative results on semialgebraic sets and the first single exponential algorithm computing their first Betti number. An index of notation has also been added.
 Reviews

From the reviews:
"The monograph gives a selfcontained detailed exposition of the algorithmic real algebraic geometry. ... In general, the monograph is well written and will be useful both for beginners and for advanced readers, who work in real algebraic geometry or apply its methods in other fields."
Eugenii I. Shustin, Zbl. MATH 1031.14028
"... The book under review gives a selfcontained account of some of the more recent and important algorithms arising in RAG [real algebraic geometry]. ... This material has mostly appeared in other sources; however, it is very nice to have it all in one book. ...the book is wonderful reference for algorithms in RAG, for the expert and nonexpert alike."
V.Powers, Mathematical Reviews Clippings from Issue 2004g
From the reviews of the second edition:
"‘Real root counting problem’ is one of the main problems under consideration in Algorithms in Real Algebraic Geometry … . the authors have posted an interactive version of the book on each of their websites. The book attempts to be selfcontained and … the authors succeed … . Basu, Pollack, and Roy have written a detailed book with quite a few examples and … bibliographic references. … The websites also contain implementations of several of the algorithms … which this reviewer found particularly illuminating." (Darren Glass, MathDL, January, 2007)
"Algorithms in Real Algebraic Geometry … provides a selfcontained treatment of some of the important classical and modern results in semialgebraic geometry, many authored by some subset of the trio Basu, Pollack, and Roy. … The authors have clearly done a tremendous service by providing a selfcontained and surprisingly complete source for the foundations of algorithmic real algebraic geometry. They have also organized their material in a way that can be reasonably taught to graduate students." (J. Maurice Rojas, Foundations of computational Mathematics, Issue 8, 2008)
 Table of contents (17 chapters)


Introduction
Pages 110

Algebraically Closed Fields
Pages 1127

Real Closed Fields
Pages 2982

SemiAlgebraic Sets
Pages 8399

Algebra
Pages 101157

Table of contents (17 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Algorithms in Real Algebraic Geometry
 Authors

 Saugata Basu
 Richard Pollack
 MarieFrançoise Roy
 Series Title
 Algorithms and Computation in Mathematics
 Series Volume
 10
 Copyright
 2006
 Publisher
 SpringerVerlag Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
 eBook ISBN
 9783540330998
 DOI
 10.1007/3540330992
 Hardcover ISBN
 9783540330981
 Softcover ISBN
 9783642069642
 Series ISSN
 14311550
 Edition Number
 2
 Number of Pages
 X, 662
 Topics