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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... called a cylindrical decom- position : a semi - algebraic set is decomposed into a finite number of pieces , each homeomorphic to an open cube . A finer decomposition discussed later in the chapter provides a stratification , i.e. a ...
... called the algebraic closure of the field ( see section 2 , chapter 5 of [ 83 ] ) . A typical example of an algebraically closed field is the field C of complex numbers . We study the sets of points which are the common zeros of a ...
... called its prenex normal form ( see Section 10 , Chapter 1 of [ 94 ] ) . The variables X1 , ... , Xm are called bound variables . If the formula has no free variables , i.e. Free ( Ø ) = Ø , then it is called a sentence , and it is ...
... called Posgcd ( P ) , which is a finite set containing all the possible greatest common divisors of Py which can occur as y ranges over Ck . We define it as a set of pairs ( G , C ) where G Є D [ Y1 , ......... , Yk ] [ X ] and C is a ...
... called the Lefschetz Principle . Theorem 1.27 ( Lefschetz principle ) . Suppose that C ' is an algebraically closed field which contains the algebraically closed field C. If is a sentence in the language of fields with coefficients in C ...
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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