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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... univariate polynomial does not always have real roots, a very natural algorithmic problem, is to design a method to ... polynomials (P,Q) have a greatest common divisor by giving a finite procedure for constructing the greatest common ...
... univariate and multivariate polynomials to linear algebra, determinants and quadratic forms. A general theme is to express some properties of univariate polynomials by the vanishing of specific polynomial expressions in their ...
... polynomial system solving. We give a few results about Gröbner bases, and explain the technique of rational univariate representation. Since our techniques in the following chapters involve infinitesimal deformations, we also indicate ...
... polynomials with coefficients in an algebraically closed field C. A field C is algebraically closed if any non-constant univariate polynomial P(X) with coefficients in C has a root in C, i.e. there exists x∈C such that P(x)=0. Every ...
... univariate situation with parameters. Viewing a univariate algorithm parametrically to obtain a multivariate ... polynomials in P and Q as polynomials in the single variable X with the variables (Y1, , Yk) appearing as parameters. For a ...
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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