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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... variables. More recent results bounding the individual Betti numbers of sign conditions defined by a family of polynomials on an algebraic set are described. These results involve a combinatorial part, depending on the number of ...
... variables, using subresul- tants. Cylindrical decomposition has numerous applications among which are: deciding the truth of a sentence, eliminating quantifiers, computing a stratification, and computing topological information of ...
... variables. Using the pseudo-critical points introduced in Chapter 12 and perturbation methods to obtain polynomials ... variables but rather in the number of blocks of variables appearing in the formula where the blocks of variables are ...
... variables , and bound variables , i.e. quantified variables . More precisely , let D be a subring of C. We define the language of fields with coefficients in D as follows . An atom is P = 0 or P0 , where P is a polynomial in D [ X1 ...
... variables contained in { Y1 , ... , Yk } , denoted Reali ( p , Ck ) , is the set of y Ck such that ( y ) is true . It is defined by induction on the construction of the formula , starting from atoms : Reali ( P = 0 , Ck ) Reali ( P +0 ...
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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