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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 semi-algebraic set appear frequently in many areas of science and engineering. In this textbook the main ideas and techniques presented form a coherent and rich body of knowledge.
Mathematicians will find relevant information about the algorithmic aspects. 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.
This second edition contains several recent results, on discriminants of symmetric matrices, real root isolation, global optimization, quantitative results on semi-algebraic sets and the first single exponential algorithm computing their first Betti number.
Mathematicians will find relevant information about the algorithmic aspects. 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.
This second edition contains several recent results, on discriminants of symmetric matrices, real root isolation, global optimization, quantitative results on semi-algebraic sets and the first single exponential algorithm computing their first Betti number.
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 ofcomputational Mathematics, Issue 8, 2008)
"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 ofcomputational Mathematics, Issue 8, 2008)