Algorithms in Real Algebraic GeometrySpringer Science & Business Media, 9. mar. 2013 - 602 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, or deciding whether two points belong in the same connected component of a semi-algebraic set occur in many contexts. In this first-ever graduate textbook on the algorithmic aspects of real algebraic geometry, the main ideas and techniques presented form a coherent and rich body of knowledge, linked to many areas of mathematics and computing. Mathematicians already aware of real algebraic geometry will find relevant information about the algorithmic aspects, and 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. |
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... proof is very elementary and uses nothing but a para- metric version of the euclidean remainder sequence . In Chapter 2 , the basic geometric objects are the semi - algebraic sets which constitute our main ob- jects of interest in this ...
... proof uses the extended signed remainder sequence defined as follows : let ( So , Uo , Vo ) = ( P , 1,0 ) , ( S1 ... Proof of Proposition 1.10 : It is easy to verify by induction on i that , for 0 ≤ i ≤ k + 1 , Si = U¿P + ViQ . Let di ...
... Proof : By Theorem 1.23 , there is a quantifier free formula V which is C- equivalent to P. It follows from the proof of Theorem 1.21 that is C'- equivalent to § as well . Notice , too , that since V is a sentence , I is a boolean ...
... Proof : Identify the coefficient of Xi on both sides of ( X - X1 ) ··· ( X - Xk ) = X + C1Xk - 1 + ··· + Ck . ... Proposition 2.18 . Let K be a field and let Q ( X1 , ... , Xk ) E K [ X1 ,. Xk ] be symmetric . There exists a polynomial ...
... proof is over . Otherwise , the leading monomial with respect to Q1 the graded lexicographical ordering of Q1 is strictly smaller than Xa1 ..Хак , and it is possible to iterate the construction with Q1 . Since there is no infinite ...
Indhold
1 | |
Algebra | 91 |
པ Decomposition of SemiAlgebraic Sets | 137 |
Elements of Topology | 173 |
Quantitative Semialgebraic Geometry | 201 |
Complexity of Basic Algorithms | 241 |
Cauchy Index and Applications | 283 |
Real Roots 321 | 320 |
Cylindrical Decomposition Algorithm 421 | 420 |
Existential Theory of the Reals | 465 |
Quantifier Elimination | 493 |
Computing Roadmaps and Connected Components | 522 |
Computing Roadmaps and Connected Components | 549 |
References 587 | 586 |
Index | 595 |
Polynomial System Solving | 365 |
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Algorithms in Real Algebraic Geometry Saugata Basu,Richard Pollack,Marie-Françoise Coste-Roy Begrænset visning - 2007 |
Algorithms in Real Algebraic Geometry Saugata Basu,Richard Pollack,Marie-Françoise Roy Begrænset visning - 2003 |
Algorithms in Real Algebraic Geometry Saugata Basu,Richard Pollack,Marie-Françoise Roy Begrænset visning - 2006 |