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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... infinitesimal elements and can be described geometrically as germs of semi-algebraic functions, and algebraically as algebraic Puiseux series. The real closed field of algebraic Puiseux series plays a key role in the complexity results ...
... infinitesimal deformations, we also indicate how to compute the limit of the bounded solutions of a polynomial system when the deformation parameters tend to zero. As a consequence, using the ideas of the critical point method described ...
... infinitesimal over F if it is a positive element smaller than any positive f G F. The element f G F' is unbounded over F if it is a positive element greater than any positive f G F. Notation 2.5. [Order 0+] Let F be an ordered field and ...
... infinitesimal elements over F, such as ε. The field also contains elements which are unbounded over F such as 1/ε. □ Exercise 2.4. Show that 0+ is an order on F(ε) and that it is the only order in which ε is infinitesimal over F. We ...
... infinitesimal elements. We shall see at the end of this chapter an example of a non-archimedean real closed field when we study the field of Puiseux series. 2.2 Real Root Counting Although we have a very simple 2.1 Ordered, Real and ...
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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