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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... follows from Hilbert's Nullstellensatz: a system of polynomial equations has a finite number of solutions in an ... follow from these decompositions. Among these are: − a semi-algebraic set has a finite number of connected components ...
... follows. An atom is P = 0 or P 0, where P is a polynomial in D[X1, ,Xk]. We define simultaneously the formulas and Free(Φ), the set of free variables of a formula Φ, as follows − an atom P = 0 or P 0, where P is a polynomial in D[X1 ...
Saugata Basu, Richard Pollack, Marie-Françoise Coste-Roy. It follows immediately from the definitions that: Proposition 1.5. Let P G KlX] and Q Q KlX], not both zero. Then PQ/G is a least common multiple ofP and Q. Corollary 1.6. deg(1 ...
... follows by induction on the number of quantifiers. □ Corollary 1.24. Let Φ(Y) be a formula in the language offields with coefficients in C. The set {y ∈Ck|Φ(y)} is constructible. Corollary 1.25. A subset of C defined by aformula in ...
... follows. Theorem 2.11. If R is a field then the following properties are equivalent: a) R is real closed. b) R[i]=R[T]/(T2+ 1) is an algebraically closed field. c) R has the intermediate value property. d) R is a real field that has no ...
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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Algorithms in Real Algebraic Geometry Saugata Basu,Richard Pollack,Marie-Françoise Roy Begrænset visning - 2003 |
Algorithms in Real Algebraic Geometry Saugata Basu,Richard Pollack,Marie-Françoise Roy Begrænset visning - 2006 |
Algorithms in Real Algebraic Geometry Saugata Basu,Richard Pollack,Marie-Françoise Coste-Roy Begrænset visning - 2013 |