Algorithms in Real Algebraic GeometrySpringer Science & Business Media, 21. apr. 2007 - 662 sider 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. |
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... . . . Decomposition of Semi-Algebraic Sets . . . . . . . . . . . . . . 5.1 Cylindrical Decomposition . . . . . . . . . . . . . . . . . . . . . . 5.2 Semi-algebraically Connected Components . . . . . . . . . . . 168 5.3 Dimension ...
... . 557 14.5 Dimension of Semi-algebraic Sets . . . . . . . . . . . . . . . . . 558 14.6 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . 562 15 Computing Roadmaps and Connected Components of Algebraic Sets ...
... algebraically closed case. Namely, the geometric statement 'an algebraic variety (the common ... Semi-algebraic sets are defined by a finite number of polynomial ... connected components each of which is semi-algebraic, − algebraic sets ...
... semi-algebraic sets, the first of them being the number of connected components of the set. We also define the Euler-Poincaré characteristic, which is a significant topological invariant of algebraic and semialgebraic sets. Chapter 7 ...
... algebraic curve. The huge degree bounds (doubly exponential in the number of variables) output by the cylindrical decomposition method give estimates on the number of connected components of semi-algebraic sets which are much worse than ...
Indhold
1 | |
11 | |
29 | |
SemiAlgebraic Sets | 83 |
4 | 100 |
Decomposition of SemiAlgebraic Sets | 159 |
6 | 195 |
Quantitative Semialgebraic Geometry | 237 |
Interval | 330 |
Existential Theory of the Reals | 505 |
Quantifier Elimination | 533 |
Computing Roadmaps and Connected Components of Alge | 563 |
Computing Roadmaps and Connected Components of Semi | 593 |
References | 635 |
132 | 641 |
Index of Notation 645 | 644 |
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