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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... Notation 1.1. [Zero set] If P is a finite subset of C[X1, ,Xk] we write the set of zeros of P in Ck as Zer(P,Ck)={x∈CkF ∧ P∈P P(x)=0}. These are the algebraic subsets of Ck. The set Ck is algebraic since Ck= Zer({0},Ck). □ Exercise ...
... notation. We get rid of denominators appearing in the remainders through the notion of signed pseudo-remainders. Let P : apXp+~-+a0GD[X], Q : bqXq+~-+b0GD[X], where D is a subring of C. Note that the only denominators occurring in the ...
... Notation 2.5. [Order 0+] Let F be an ordered field and e a variable. There is one and only one order on F(e), denoted 0+, such that e is infinitesimal over F. If P(e) I apep+apI1eIV1 + + am+1em+1 +am em with am 0, then P(ε) > 0 in this ...
... 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(p). U Proposition 2.27. [Basic Thom's Lemma] Let P be a univariate polynomial of degree p and let a be a sign ...
... 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 if a0a1<0 Var(a1,,ap) if a0a1>0 This ...
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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