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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... elements of C. A greatest common divisor of P and Q , denoted gcd ( P , Q ) , is a poly- nomial G Є K [ X ] such ... element in K \ { 0 } . Two polynomials are coprime is their greatest common divisor is an element of K \ { 0 } . We now ...
... elements over F , such as ɛ . The field also contains elements which are unbounded over F such as 1 / ɛ . Exercise 2.6 . Show that this is an order on F ( ɛ ) and that it is the only order in which ε is infinitesimal over F. We define ...
... elements of F and by F ( 2 ) the set of sums of squares of elements of F. Clearly , F ( 2 ) is a cone contained in every cone of F. A field F is a real field if −1 [ F ( 2 ) . Note that a real field must have characteristic 0 ...
... elements of a field K and P = ( X - X1 ) ... , ( X - Xk ) = X + C1Xk - 1 + ... + Ck , then C1 = ( -1 ) 1E . Ci ... element and the graded lexicographical ordering order is com- patible with multiplication . Note that the set of monomials ...
... elements of the real closure are algebraic over F ( i.e. satisfy an equation with coefficients in F ) . We refer the reader to [ 25 ] for these results . Exercise 2.31 . If F is contained in a real closed field R , the real closure of F ...
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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