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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... Sign Determination . . . . . . . . . . . . . . . . . . . . . . . . . . 383 10.4 Roots in a Real Closed Field . . . . . . . . . . . . . . . . . . . . 397 10.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . 401 ...
... signature of related quadratic forms, as well as an application to counting complex roots in a half plane, useful in ... sign determination and characterizing real roots, without approximating the roots. In Chapter 11, we describe an ...
... determining whether a polynomial P ∈C[X] has a root in C (namely, if and only if deg(P) 0), it is much more difficult ... Signs and the Budan-Fourier Theorem Notation 2.32. [Sign variations] The number of sign variations, Var(a), in a ...
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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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