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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... union of strata of lower dimension . We also describe how to triangulate a closed and bounded semi - algebraic set . Various other finiteness results about semi - algebraic sets follow from these decompositions . Among these are : a ...
... union of basic constructible sets . The principal goal of this chapter is to prove that the projection from Ck + 1 to C that is defined by " forgetting " the last coordinate maps constructible sets to constructible sets . For this ...
... union of those parts where Sy Ø . In fact , the decision method is the same ( is uniform ) for all y in any given part . Thus each part is a constructible set and consequently ( S ) is constructible being the union of finitely many ...
... by polynomials with coefficients in D , its projection to Ck is a constructible set defined by polyno- mials with coefficients in D. Proof : Since every constructible set is a finite union 1.3 Projection Theorem for Constructible Sets 21.
... union of basic constructible sets it is sufficient to prove that the projection of a basic constructible set is constructible . Suppose that the basic constructible set S in Ck + 1 is X { ( y , x ) = Ck × C / P ( y , x ) = 0 ^ ^ Q ( y ...
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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