Algorithms in Real Algebraic Geometry / Edition 2

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This is the first graduate textbook on the algorithmic aspects of real algebraic geometry. 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 and other relevant topics.

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Editorial Reviews

From the Publisher
"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 zählen beispielsweise die Frage nach der Existenz reeller Lösungen einer (durch polynomiale Gleichungen und Ungleichungen) bestimmten semialgebraischen Menge oder die Frage, ob zwei Punkte zur gleichen Zusammenhangskomponente einer gegebenen semialgebraischen Menge gehören. [...] Eine der Herausforderungen kommt daher, dass hierbei eine Reihe von Teilgebieten der Mathematik und Informatik wie Topologie, algebraische Geometrie, Computeralgebra, Komplexitätstheorie sowie der Entwurf effizienter Algorithmen eng miteinander verzahnt sind und deshalb die Literatur sehr verstreut war. In genau diese Lücke möchte das vorliegende Buch stoßen - und dieses Unterfangen ist den Autoren in beeindruckender Weise gelungen! [...] Das Buch bietet eine sehr zeitgemäße, gelungene Darstellung klassischen sowie aktuellen Materials zu algorithmischen Fragen der reellen algebraischen Geometrie, die in dieser Breite bisher nicht verfügbar war. Besonders auffällig ist die erfolgreiche Absicht der Autoren, eine kohärente und vor allem in sich geschlossene Darstellung zu liefern, die die verschiedenen beteiligten mathematischen Teilgebiete umfassend berücksichtigt. Aufgrund dieser Darstellungsweise bietet das Bcuh zahlreiche Einstiegs- und Verwendungsmöglichkeiten, 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

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Product Details

  • ISBN-13: 9783540330981
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 8/21/2006
  • Series: Algorithms and Computation in Mathematics Series, #10
  • Edition description: 2nd ed. 2006
  • Edition number: 2
  • Pages: 662
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.50 (d)

Table of Contents

1 Algebraically closed fields 11
2 Real closed fields 29
3 Semi-algebraic sets 83
4 Algebra 101
5 Decomposition of semi-algebraic sets 159
6 Elements of topology 195
7 Quantitative semi-algebraic geometry 237
8 Complexity of basic algorithms 281
9 Cauchy index and applications 323
10 Real roots 351
11 Cylindrical decomposition algorithm 403
12 Polynomial system solving 445
13 Existential theory of the reals 505
14 Quantifier elimination 533
15 Computing roadmaps and connected components of algebraic sets 563
16 Computing roadmaps and connected components of semi-algebraic sets 593
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