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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... Projection Theorem for Constructible Sets . . . . . . . . . . . 20 1.4 Quantifier Elimination and the Transfer ... Projection Theorem for Algebraic Sets . . . . . . . . . . . . . . 57 2.4 Projection Theorem for Semi-Algebraic Sets ...
... projection of a constructible set is constructible. The proof is very elementary and uses nothing but a parametric version of the euclidean remainder sequence. In Chapter 2, the basic geometric objects are the semi-algebraic sets which ...
... projection of an algebraic set in affine space is constructible. Considering projective space allows an even more satisfactory result: the projection of an algebraic set in projective space is algebraic. This result appears here as a ...
... projection on a line of a smooth hypersurface are precisely the places where a change in topology occurs in the part of the hypersurface inside a half space defined by a hyperplane orthogonal to the line. Counting these critical points ...
... projection of a constructible set is constructible. Section 1.1 is devoted to definitions. The main technique used for proving the projection theorem in Section 1.3 is the remainder sequence defined in Section 1.2 and, for the case ...
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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