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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... sets. These are the subsets of Cn which are defined by a finite number of polynomial equations (P = 0) and inequations (P 0). We prove that the projection of a constructible set is constructible. The proof is very elementary and uses ...
... polynomials are the classical resultant and its generalization to subresultant coefficients. The vanishing of these ... set in affine space is constructible. Considering projective space allows an even more satisfactory result: the projection ...
... set. We also define the Euler-Poincaré characteristic, which is a significant topological invariant of algebraic and ... polynomials on an algebraic set are described. These results involve a combinatorial part, depending on the number ...
... polynomials, which provide useful constructions in CAD (Computer Aided ... set. Since we deal with arbitrary algebraic sets which are not necessarily ... set with complexity polynomial in the degree and exponential in the number of ...
... set taking advantage of the (possibly low) dimension of the algebraic set. We also compute the Euler-Poincaré characteristic of sign conditions defined by a set of polynomials. The complexity results obtained are quite satisfactory 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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