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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... closed fields (such as the field of complex numbers C) and real closed fields (such as the field of real numbers R). The concept of a real closed field was first introduced by Artin and Schreier in the 1920's and was used for their ...
... closed fields and real closed fields. In Chapter 1, the basic geometric objects are constructible sets. These are the subsets of Cn which are defined by a finite number of polynomial equations (P = 0) and inequations (P 0). We prove ...
... real closed field, isolation techniques are no longer possible. We prove that a root of a polynomial can be uniquely described by sign conditions on the derivatives of this polynomial, and we describe a different method for performing ...
... 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 ... Real Closed Fields Real closed fields are fields which 1.5 Bibliographical Notes 27 Bibliographical Notes.
Saugata Basu, Richard Pollack, Marie-Françoise Coste-Roy. 2. Real. Closed. Fields. Real closed fields are fields which share the algebraic properties of the field of real numbers. In Section 2.1, we define ordered, real and real closed ...
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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