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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... Chapter 8. The first part of the book (Chapters 1 through 7) consists primarily of the mathematical background ... Chapter 1 and Chapter 2, we study algebraically closed fields (such as the field of complex numbers C) and real closed ...
Saugata Basu, Richard Pollack, Marie-Françoise Coste-Roy. Chapters 1 and 2 describe an interplay between geometry and logic for algebraically closed fields and real closed fields. In Chapter 1, the basic geometric objects are ...
... Chapter 1, the 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 ...
... Chapter 7 presents basic results of Morse theory and proves the classical Oleinik-Petrovsky-Thom-Milnor bounds on the sum of the Betti numbers of an algebraic set of a given degree. The basic technique for these results is the critical ...
... Chapter 11, we describe an algorithm for computing the cylindrical decomposition which had been already studied in Chapter 5. The basic idea of this algorithm is to successively eliminate variables, using subresultants. Cylindrical ...
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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