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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... Algebraic Sets ... 402 11.7 Computing the Euler - Poincaré Characteristic of an Algebraic Set .... 414 11.8 Bibliographical Notes 419 12 Cylindrical Decomposition Algorithm 12.1 Computing the Cylindrical Decomposition 12.1.1 Contents VII.
... algebraic Sets 517 • 14.5 Bibliographical Notes 521 15 Computing Roadmaps and Connected Components of Algebraic Sets . 523 15.1 Pseudo - critical Values and Connectedness 524 15.2 Roadmap of an Algebraic Set 526 15.3 Computing Connected ...
... algebraic sets which constitute our main ob- jects of interest in this book . These are the subsets of R " that are ... algebraic set is semi - algebraic . The proof , though more complicated than that for the algebraically closed case ...
... algebraic set in affine space is constructible . Considering projective space allows an even more satisfactory result : the pro- jection of an algebraic set in projective space is algebraic . This result appears here as a consequence of ...
... algebraic sets , the first of them being the num- ber of connected components of the set . We also define the Euler - Poincaré characteristic , which is a significant topological invariant of algebraic and semi - algebraic sets ...
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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