Algorithms in Real Algebraic Geometry
Springer 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.
Resultater 1-5 af 84
We study the sets of points which are the common zeros of a finite family of
polynomials. If D is a ring, we denote by D[X1, , Xk] the polynomials in k variables
X1, ,Xk with coefficients in D. Notation 1.1. [Zero set] If P is a finite subset of C[X1,
Let P be a non-zero polynomial P=apXp++a1X+a0∈D[X] with ap 0. We denote
the degree of P, which is p, by deg (P). By convention, the degree of the zero
polynomial is defined to be −∞. If P is non-zero, we write cofj(P) = aj for the
Let P G KlX] and Q Q KlX], not both zero. Then PQ/G is a least common multiple
ofP and Q. Corollary 1.6. deg(1<1m(P,Q)): deg(P) + des(Q) — des(s<rd(P. Q))We
now prove that greatest common divisors and least common multiple exist by ...
... PRem(P, Q) G D[X]. The even exponent is useful in Chapter 2 and later when
we deal with signs. Notation 1.16. [Truncation] Let Q I bq Xq + + b0 G D[X]. We
define for 0 g i g q, the truncation of Q at i by The set of truncations of a non-zero ...
2.1. Ordered,. Real. and. Real. Closed. Fields. Before defining ordered fields, we
prove a few useful properties of fields of characteristic zero. Let K be a field of
characteristic zero. The derivative of a polynomial P=a pXp+ +aiXi+ ...
Hvad folk siger - Skriv en anmeldelse
Existential Theory of the Reals
Computing Roadmaps and Connected Components of Alge
Computing Roadmaps and Connected Components of Semi
Index of Notation 645
Complexity of Basic Algorithms