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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... non constant polynomial with coefficients in C has a root in C , which is expressed by \ PЄC [ X ] deg ( P ) > 0 , ⇒ X € CP ( X ) = 0 . This expression is not a sentence of ... non - zero polynomial with ap 14 1 Algebraically Closed Fields.
... non - zero polynomial with ap +0 . P = apX2 + ··· + a1 X + ao Є D [ X ] ... 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 ...
... non - zero polynomial QЄ D [ Y1 , Yk ] [ X ] , where Y1 , ... , Yk are parameters and X is the main variable , is the finite subset of D [ Y1 , ... , Yk ] [ X ] defined by S { Q } Tru ( Q ) = { if lcof ( Q ) ED or deg ( Q ) = 0 , { Q } ...
... non - zero squares of elements in C ) to P , P ' , S2 , S3 , S4 . □ Notation 1.18 . [ Degree ] For a specialization of Y = ( Y1 , ... , Yk ) to yЄ Ck , and QE DY1 , ... , Yk ] [ X ] , we denote the polynomial in C [ X ] obtained by ...
... non-archimedean real closed field is described: the field of Puiseux series. 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 ...
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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Algorithms in Real Algebraic Geometry Saugata Basu,Richard Pollack,Marie-Françoise Roy Begrænset visning - 2003 |
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