Algorithms in Real Algebraic Geometry: v.10 (Algorithms and Computation in Mathematics) - Hardcover

Basu, Saugata; Pollack, Richard; Roy, Marie-Francoise

Hardcover

ISBN 10: 3540009736 ISBN 13: 9783540009733

Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K, 2003

Synopsis

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.

About this title

Review

"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

"Gegenstand des Buches sind zentrale algorithmische Probleme der reellen algebraischen Geometrie. Hierzu zAhlen beispielsweise die Frage nach der Existenz reeller LAsungen einer (durch polynomiale Gleichungen und Ungleichungen) bestimmten semialgebraischen Menge oder die Frage, ob zwei Punkte zur gleichen Zusammenhangskomponente einer gegebenen semialgebraischen Menge gehAren. [...] Eine der Herausforderungen kommt daher, dass hierbei eine Reihe von Teilgebieten der Mathematik und Informatik wie Topologie, algebraische Geometrie, Computeralgebra, KomplexitAtstheorie sowie der Entwurf effizienter Algorithmen eng miteinander verzahnt sind und deshalb die Literatur sehr verstreut war. In genau diese LA1/4cke mAchte das vorliegende Buch stoAen - und dieses Unterfangen ist den Autoren in beeindruckender Weise gelungen! [...] Das Buch bietet eine sehr zeitgemAAe, gelungene Darstellung klassischen sowie aktuellen Materials zu algorithmischen Fragender reellen algebraischen Geometrie, die in dieser Breite bisher nicht verfA1/4gbar war. Besonders auffAllig ist die erfolgreiche Absicht der Autoren, eine kohArente und vor allem in sich geschlossene Darstellung zu liefern, die die verschiedenen beteiligten mathematischen Teilgebiete umfassend berA1/4cksichtigt. Aufgrund dieser Darstellungsweise bietet das Bcuh zahlreiche Einstiegs- und VerwendungsmAglichkeiten, sowohl in Lehre und Forschung als auch als Nachschlagewerk. Es wird sich schnell als Standardwerk zu dem behandelten Themenkreis etablieren."

T.Theobald, Jahresberichte 107, Band (2005) Heft 1

From the Back Cover

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.

Edition details

• Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
• Publication date: 2003
• Language: English
• ISBN 10: 3540009736
• ISBN 13: 9783540009733
• Binding: Hardcover
• Number of pages: 610