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. 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 ivaluable parts of it, to undergraduate students. 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 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. This revised second edition contains several recent results, notably 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. An index of notation has also been added front-matter......Page 1 1Introduction......Page 9 2Algebraically Closed Fields......Page 19 3Real Closed Fields......Page 36 4Semi-Algebraic Sets......Page 90 5Algebra......Page 107 6Decomposition of Semi-Algebraic Sets......Page 164 7Elements of Topology......Page 200 8Quantitative Semi-algebraic Geometry......Page 242 9Complexity of Basic Algorithms......Page 286 10Cauchy Index and Applications......Page 328 11Real Roots......Page 356 12Cylindrical Decomposition Algorithm......Page 407 13Polynomial System Solving......Page 449 14Existential Theory of the Reals......Page 508 15Quantifier Elimination......Page 536 16Computing Roadmaps and Connected Components of Algebraic Sets......Page 566 17Computing Roadmaps and Connected Components of Semi-algebraic Sets......Page 596 back-matter......Page 638 "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." Since a real univariate polynomial does not always have real roots, a very natural algorithmic problem, is to design a method to count the number of real roots of a given polynomial (and thus decide whether it has any). Saugata Basu, Richard Pollack, Marie-françoise Roy. Includes Bibliographical References And Index.
