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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... ordered set is the power set 24 = { B | BC A } , the binary relation being the inclusion between subsets of A. A totally ordered set is a partially ordered set ( A , ≤ ) with the additional property that every two elements a , b Є A ...
... ordered field -1 < 0 . ( b ) Prove that an ordered field has characteristic zero . ( c ) Prove the law of trichotomy in an ordered field : for every a in the field , exactly one of a < 0 , a = 0 , a > 0 holds . Exercise 2.3 . Show that ...
... ordered fields . The element fЄ F ' is infinitesimal over F if it is a positive element smaller than any positive ƒ ... ordered field and ɛ a variable . There is one and only one order on F ( ɛ ) , denoted 0+ , such that ɛ is ...
... ordered . ( iv ) For every X1 , Xn in F n = 1x ← 0 = ¿ x = Xn = 0 . i = 1 Proof : ( i ) ⇒ ( ii ) , since in a real field F , ΣF ( 2 ) is a proper cone . ( ii ) ( iii ) by Lemma 2.10 . x1 n ( iii ) ⇒ ( iv ) , since in an ordered field ...
... ordered field whose positive cone is the set of squares R ( 2 ) and such that every polynomial in R [ X ] of odd degree has a root in R. Note that the condition that the positive cone of a real closed field R is R ( 2 ) means that R has ...
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 |