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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... First Properties 1.2 Euclidean Division and Greatest Common Divisor 1.3 Projection Theorem for Constructible Sets 1.4 Quantifier Elimination and the Transfer Principle 1.5 Bibliographical Notes 1 11 11 14 2 Real Closed Fields 2.1 ...
... First Properties . . . . . . . . . . . . . . 221 6.2.2 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.3 Homology of Certain Locally Closed Semi-Algebraic Sets . . 226 6.3.1 Homology of Closed Semi-algebraic Sets ...
... first part of the book ( Chapters 1 through 7 ) consists primarily of the mathematical background needed for the second part . Much of this back- ground is already known and has appeared in various texts . Since these results come from ...
... first Betti number of semi-algebraic sets. Moreover, it gives an efficient algorithm for computing semi-algebraic descriptions of the connected components of a semi-algebraic set in single exponential time. 1 Warning This book is ...
... first time, and do not describe the full history and bibliography. We also list below the references containing the material we have used directly. 3 Existing implementations In terms of existing implementation of the algo- rithms ...
Indhold
1 | |
43 | |
94 | 77 |
3 | 83 |
Algebra | 100 |
པ | 159 |
213 | 183 |
Elements of Topology | 195 |
Real Roots | 365 |
Cylindrical Decomposition Algorithm | 402 |
Polynomial System Solving | 445 |
Existential Theory of the Reals | 505 |
Quantifier Elimination | 533 |
Computing Roadmaps and Connected Components of Alge | 564 |
Computing Roadmaps and Connected Components of Semi | 593 |
References | 635 |
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