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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... lemma is clear: Lemma 1.14. If'P, Q are two finite subsets of D[X], then there is an at G C such that (A P<I)O>A( /\ who) P6P QGQ if and only if degacdacdw). H on) #degacdm). QGQ where d is any integer greater than deg(gcd(P)). Note ...
... Lemma 1.19. The Reali(CL) partition C'“. Moreover, yGReali(CL) implies that the signed remainder sequence of Py and ... lemma is true: Lemma 1.20. For all y G Ck, there exists one 1.3 Projection Theorem for Constructible Sets 23.
... Lemma 1.19 that Lemma 1.20 holds. Example 1.21. Returning to Example 1.17, and using the corresponding notation, the elements of posgcd(P, P') are (after removing obviously irrelevant factors), 9'13)' aIOAsIOA5IOL aI0AsI0A5I0 ...
... lemma. Lemma 2.9. Let C be a proper cone of F. If −a∈C, then C[a]={x+ayFx,y∈C} is a proper cone of F. Proof: Suppose −1= x+ ay with x, y ∈C. If y =0 we have −1 ∈C which is impossible. If y 0 then −a=(1/y2)y (1 +x)∈C, which is ...
... lemmas. Lemma 2.41. The polynomial X −x is normal if only if x⩽0. Proof: Follows immediately from the definition of a normal polynomial. □ Lemma 2.42. A quadratic monic polynomial A with complex conjugate roots is normal if and only ...
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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