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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... Rational Functions 302 9.2.2 Signature of Hankel Quadratic Forms 305 9.3 Number of Complex Roots with Negative Real Part .. 9.4 Bibliographical Notes 313 .319 10 Real Roots ... 321 10.1 Bounds on Roots 321 10.2 Isolating Real Roots 10.3 ...
... rational univariate repre- sentation . Since our techniques in the following chapters involve infinitesimal deformations , we also indicate how to compute the limit of the bounded solu- tions of a polynomial system when the deformation ...
... rational numbers . Let P be a non - zero polynomial with ap +0 . PapXP + ... + a1X + ao € D [ X ] = We denote the degree of P , which is p , by deg ( P ) . By convention , the degree of the zero polynomial is defined to be -∞o . If P ...
... rational functions R ( X ) . For this purpose , we need a definition : Let FC F ' be two ordered fields . The element fЄ F ' is infinitesimal over F if it is a positive element smaller than any positive ƒ € F. The element ƒ € F ' is ...
... rational numbers . Let R be a maximal real subfield of C. The field R is real closed since it has no nontrivial real algebraic extension contained in C ( see Theorem 2.11 ) . Note that CR cannot contain a t which is transcendental over ...
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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