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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... contained , coherent ex- position of these topics . In the first and second chapters , we study algebraically closed fields ( such as the field of complex numbers C ) and real closed fields ( such as the field of real numbers R ) . The ...
... contained in every cone of F. A field F is a real field if −1 [ F ( 2 ) . Note that a real field must have characteristic 0 . Exercise 2.9 . Prove that a real field has characteristic 0 . Show that an ordered field must be a real field ...
... contained in a real closed field R , the real closure of F consists of the elements of R which are algebraic over F ... contained in C ( see Theorem 2.11 ) . Note that CR cannot contain a t which is transcendental over R since otherwise ...
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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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