Algorithms in Real Algebraic Geometry

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Algorithms and Computation in Mathematics 10

ISBN: 3540330984
ISBN 13: 9783540330981
Autor: Basu, Saugata/Pollack, Richard/Coste-Roy, Marie-Françoise
Verlag: Springer Verlag GmbH
Umfang: x, 662 S.
Erscheinungsdatum: 06.07.2006
Auflage: 2/2006
Format: 4 x 24 x 16.7
Gewicht: 1117 g
Produktform: Gebunden/Hardback
Einband: GEB

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 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 semi-algebraic sets and the first single exponential algorithm computing their first Betti number. An index of notation has also been added.

Artikelnummer: 409009 Kategorie:

Beschreibung

InhaltsangabeAlgebraically 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.

Inhaltsverzeichnis

Introduction.- 1 Algebraically Closed Fields.- 2 Real Closed Fields.- 3 Semi-Algebraic Sets.- 4 Algebra.- 5 Decomposition of Semi-Algebraic Sets.- 6 Elements of Topology.- 7 Quantitative Semi-Algebraic Geometry.- 8 Complexity of Basic Algorithms.- 9 Cauchy Index and Applications.- 10 Real Roots.- 11 Cylindrical Decomposition Algorithm.- 12 Polynomial System Solving.- 13 Existential Theory of the Reals.- 14 Quantifier Elimination.- 15 Computing Roadmaps and Connected Components of Algebraic Sets.- 16 Computing Roadmaps and Connected Components of Semi-Algebraic Sets.- References.- Index of Notation.- Index.

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