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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... Puiseux Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.7 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . 81 3 Semi-Algebraic Sets ........................... 83 3.1 Topology . . . .
... Puiseux series. The real closed field of algebraic Puiseux series plays a key role in the complexity results of Chapters 13 to 16. Chapter 4 describes several algebraic results, relating in various ways properties of univariate and ...
... Puiseux series. 2.1. Ordered,. Real. and. Real. Closed. Fields. Before defining ordered fields, we prove a few useful properties of fields of characteristic zero. Let K be a field of characteristic zero. The derivative of a polynomial P=a ...
... Puiseux series. 2.2 Real Root Counting Although we have a very simple 2.1 Ordered, Real and Real Closed Fields 43.
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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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