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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... defined by a hyperplane orthogonal to the line. Counting these critical points using Bezout's theorem yields the Oleinik-Petrovsky-Thom-Milnor bound on the sum of the Betti numbers of an algebraic hypersurface, which is polynomial in ...
... defined in Section 1.2 and, for the case where the coefficients have parameters, the tree of possible pseudo-remainder sequences. Several important applications of logical nature of the projection theorem are given in Section 1.4. 1.1.
... defined by “forgetting" the last coordinate maps constructible sets to constructible sets. For this, since ... define the language of fields by describing the formulas of this language. The formulas are built starting with atoms, which ...
... defined by polynomials with coefficients in D, its projection to Ck is a constructible set defined by polynomials with coefficients in D. Proof: Since every constructible set is a finite union of basic constructible sets it is ...
... defined by aformula in the language offields with coefficients in C is a finite set or the complement of a finite set. Proof: By Corollary 1.24, a subset of C defined by a formula in the language of fields with coefficients in C is ...
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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