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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... denote the degree of P, which is p, by deg (P). By convention, the degree of the zero polynomial is defined to be ... denoted Rem(P, Q), is the unique polynomial R ∈ K[X] of degree smaller than the degree of Q such that P = A Q + R with ...
... denote the polynomial in C[X] obtained by substituting y for Y by Qy. Given Q C D[Y1, ..., Yk] [X], we define Qy C C ... denote by BL the unique path from the root of TRems(P, Q) to the leaf L. If N is a node in BL which is not a leaf ...
Saugata Basu, Richard Pollack, Marie-Françoise Coste-Roy. Let C denote an algebraically closed field and C an algebraically closed field containing C. Given a constructible set S in Ck, the extension of S to C, denoted Ext(S, C) is the ...
... Denote by $1, the group of permutations of {1, ..., If X“ I X111 Xz'k, denote Xg'IX:(11)Xg(kk) and MaI EGGS Xg'. Prove P that every symmetric polynomial can be written as a finite sum 2 00, Ma. For iI 1, ..., k, the i-th elementary ...
Saugata Basu, Richard Pollack, Marie-Françoise Coste-Roy. We denote by Mk the set of monomials in k variables X1, , X ... denote the i-th elementary symmetric function evaluated at x1 , , xk . Since P ∈ K[X], Lemma 2.12 gives ei ∈ K. By ...
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 |