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)

• Algebraically Closed Fields.- Real Closed Fields.- Semi-Algebraic Sets.- Algebra.- Decomposition of Semi-Algebraic Sets.- Elements of Topology.- Quantitative Semi-algebraic Geometry.- Complexity of Basic Algorithms.- Cauchy Index and Applications.- Real Roots.- Cylindrical Decomposition Algorithm.- Polynomial System Solving.- Existential Theory of the Reals.- Quantifier Elimination.- Computing Roadmaps and Connected Components of Algebraic Sets.- Computing Roadmaps and Connected Components of Semi-algebraic Sets.

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)

• Algebraically Closed Fields.- Real Closed Fields.- Semi-Algebraic Sets.- Algebra.- Decomposition of Semi-Algebraic Sets.- Elements of Topology.- Quantitative Semi-algebraic Geometry.- Complexity of Basic Algorithms.- Cauchy Index and Applications.- Real Roots.- Cylindrical Decomposition Algorithm.- Polynomial System Solving.- Existential Theory of the Reals.- Quantifier Elimination.- Computing Roadmaps and Connected Components of Algebraic Sets.- Computing Roadmaps and Connected Components of Semi-algebraic Sets.

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)

• Algebraically Closed Fields.- Real Closed Fields.- Semi-Algebraic Sets.- Algebra.- Decomposition of Semi-Algebraic Sets.- Elements of Topology.- Quantitative Semi-algebraic Geometry.- Complexity of Basic Algorithms.- Cauchy Index and Applications.- Real Roots.- Cylindrical Decomposition Algorithm.- Polynomial System Solving.- Existential Theory of the Reals.- Quantifier Elimination.- Computing Roadmaps and Connected Components of Algebraic Sets.- Computing Roadmaps and Connected Components of Semi-algebraic Sets.