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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... coefficients. The discriminant of a univariate polynomial P, for example, is a polynomial in the coefficients of P which vanishes when P has a multiple root. The discriminant is intimately related to real root counting, since, for ...
... coefficients. The vanishing of these subresultant coefficients expresses the fact that the greatest common divisor of two polynomials has at least a given degree. The resultant makes possible a constructive proof of a famous theorem of ...
... coefficients have parameters, the tree of possible pseudo-remainder sequences. Several important applications of ... coefficients in an algebraically closed field C. A field C is algebraically closed if any non-constant univariate ...
... coefficients in D is C-equivalent to a a formula (Qu1X1)(QumXm) B(X1, ,Xm ,Y1, Y k ) where each Qui ∈ {∀, ∃} and ... coefficients in C has a root in C, which is expressed by ∀P ∈C[X] deg(P)>0, ∃X ∈CP(X)=0. This expression is not a ...
... 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 sufficient to prove that the ...
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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