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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... induction on i that , for 0 ≤ i ≤ k + 1 , Si = U¿P + ViQ . Let di = deg ( Si ) and note that di < di - 1 . We prove by induction that , for 1 ≤ i ≤ k deg ( Ui + 1 ) = q - di , and deg ( Vi + 1 ) = p - di . Clearly , since U2 = −1 ...
... induction on the number of quantifiers . Corollary 1.24 . Let P ( Y ) be a formula in the language of fields with coeffi- cients in C. The set { y = C ( y ) } is constructible . Corollary 1.25 . A subset of C defined by a formula in the ...
... induction on m that P has a root in R [ i ] . If m = 0 , then p is odd and P has a root in R. Suppose the result is true for m – 1. Let x1 , ... , x be the roots of P ( counted with multiplicities ) in an algebraically Хр closed field ...
... induction on p . The claim is clearly true if p = 0. Suppose that Taylor's formula holds for p - 1 : p- Xp - 1 = Σ i = 0 Then , since X = x + ( X − x ) , - ( p − 1 ) ! ( p − 1 − i ) ! i ! p - 1 Хр X2 = ( x + ( x − x ) ) Σ ( p - 1 ) ...
... induction on the degree of P. The claim is obviously true for deg ( P ) = 1. Suppose that the claim is true for every polynomial of degree < d . Since P ' = ( X − x ) μ - 1 ( μQ + ( X − x ) Q ' ) , and μQ ( x ) 0 , by induction ...
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 |