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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... Semi-Algebraic Sets . . . . . . . . . . 63 2.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.5.1 Quantifier Elimination and the Transfer Principle . . 69 2.5.2 Semi-Algebraic Functions ...
... Semi-Algebraic Sets . . 226 6.3.1 Homology of Closed Semi-algebraic Sets and of Sign Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 6.3.2 Homology of a Pair . . . . . . . . . . . . . . . . . . . . . . 228 6.3 ...
... semi-algebraic sets which constitute our main objects of interest in this book. These are the subsets of Rn that are defined by a finite number of polynomial equations (P = 0) and inequalities (P > 0). We prove that the projection of a semi ...
... semi-algebraic sets in higher dimensions. In Chapter 5 we study various decompositions of a semi-algebraic set into a finite number of simple pieces. The most basic decomposition is called a cylindrical decomposition: a semi-algebraic ...
... semi-algebraic sets provide an indication about the complexity results to be hoped for when studying various algorithmic problems related to semi-algebraic sets. The second part of the book discusses various algorithmic problems in ...
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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