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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Resultater 1-5 af 83
... coefficients of a univariate polynomial which vanishes when the polyno- mial has a multiple root . The discriminant ... coefficients of two polynomials are the classical resultant and its generalization to subresultant coefficients . The ...
... coefficients in an algebraically closed field C. A field C is algebraically closed if any non - constant univariate polyno- mial P ( X ) with coefficients in C has a root in C. That is , there is a cЄ C such that P ( c ) = 0. Every ...
... coefficients in Z and Free ( ) = Free ( V ) = { X } . ( X ‡ 0 ) are Note that the formula V is quantifier free . Moreover , and I are C - equivalent since R ( Þ , C ) = { x = C | 3y € C xy − 1 = 0 } = { xЄC | x 0 } = R ( ¥ , C ) . It ...
... coefficients in C has a root in C , which is expressed by V polynomial P deg ( P ) > 03 XP ( X ) = 0 . This expression is not a sentence of the language of fields with coefficients in C , since quantification over all polynomials is not ...
... coefficients ( a , b , c ) ) for which deg ( gcd ( P , P ' ) ) > ≥ 1. Therefore , the formula 3xP ( x ) = P ' ( x ) ... coefficients in D , its projection to Ck is a constructible set defined by polyno- mials with coefficients in D. Proof ...
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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