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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... P are linear or have the form (X —c)2+d2I (X— c—id)(X— c+id),dI0 with c, d G ... sign if and only if its discriminant b2 — 4 a c is negative. Hint: the ... P does not vanish in (a, b), then P has constant sign in the interval (a, b) ...
... P(")(r))~ B We next show that univariate polynomials over a real closed field R share some of the well known basic ... sign on [a, b] by Proposition 2.20. Then P' I (X I a)mI1 (X I b)"I1 Q1, where Q1Im(X—b)Q+n(X—a)Q+(X—a)(X—b)Q'. Thus Q1 ...
... sign condition a is Reali(o)I{xGRk| /\ sign(Q(x))Io(Q)}. QGQ The sign condition a is realizable if Reali(o) is non-empty. U Notation 2.26. [Derivatives] Let P be a univariate polynomial of degree p in R[X]. We denote by Der(P) the list P, P ...
... P(d−k+1), and, on an interval, the sign of the derivative of a polynomial determines whether it is increasing or decreasing. □ Definition 2.29. Let P ∈R[X] and σ ∈{0,1,−1}Der(P), a sign condition on the set Der(P) of derivatives of ...
... Signs and the Budan-Fourier Theorem Notation 2.32. [Sign variations] The number of sign variations, Var(a), in a sequence, a = a0, , ap, of elements in R \ {0} is defined by induction on p by: Var(a0) = 0 Var(a0,,ap) = { Var(a1,,ap)+1 ...
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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