Algorithms in Real Algebraic Geometry / Edition 2

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Overview

The algorithmic problems of real algebraic geometry such as real root counting, deciding the existence of solutions of systems of polynomial equations and inequalities, or deciding whether two points belong in the same connected component of a semi-algebraic set occur in many contexts. In this first-ever 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 self-contained the book is accessible to graduate students and even, for invaluable parts of it, to undergraduate students.

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Editorial Reviews

From the Publisher
From the reviews:

"The monograph gives a self-contained 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 self-contained 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 non-expert 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 self-contained 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 self-contained treatment of some of the important classical and modern results in semi-algebraic geometry, many authored by some subset of the trio Basu, Pollack, and Roy. … The authors have clearly done a tremendous service by providing a self-contained 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)

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Product Details

  • ISBN-13: 9783540330981
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 8/21/2006
  • Series: Algorithms and Computation in Mathematics Series , #10
  • Edition description: 2nd ed. 2006
  • Edition number: 2
  • Pages: 662
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.50 (d)

Table of Contents

1 Algebraically closed fields 11
2 Real closed fields 29
3 Semi-algebraic sets 83
4 Algebra 101
5 Decomposition of semi-algebraic sets 159
6 Elements of topology 195
7 Quantitative semi-algebraic geometry 237
8 Complexity of basic algorithms 281
9 Cauchy index and applications 323
10 Real roots 351
11 Cylindrical decomposition algorithm 403
12 Polynomial system solving 445
13 Existential theory of the reals 505
14 Quantifier elimination 533
15 Computing roadmaps and connected components of algebraic sets 563
16 Computing roadmaps and connected components of semi-algebraic sets 593
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