Algorithms in Real Algebraic GeometrySpringer Science & Business Media, 21. apr. 2007 - 662 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, finding global maxima or deciding whether two points belong in the same connected component of a semi-algebraic set appear frequently in many areas of science and engineering. In this textbook the main ideas and techniques presented form a coherent and rich body of knowledge. Mathematicians will find relevant information about the algorithmic aspects. 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. This second edition contains several recent results, on discriminants of symmetric matrices, real root isolation, global optimization, quantitative results on semi-algebraic sets and the first single exponential algorithm computing their first Betti number. |
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... Polynomial Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.4.1 Hilbert's Basis Theorem ... Systems . . . . . . . . . . . . . . . . . . . . . . 143 4.6 Multivariate Hermite's Quadratic Form . . . . . . . . . . . . . 149 ...
... Polynomial System Solving . . . . . . . . . . . . . . . . . . . . . . 445 12.1 A Few Results on Gröbner Bases ... Polynomial System . . . . . . . . 471 12.6 Finding Points in Connected Components of Algebraic Sets . 483 12.7 Triangular ...
... system of Diophantine equations has an integer solution. In Chapter 3 we ... polynomial expressions in their coefficients. The discriminant of a univariate ... polynomial expressions in the coefficients of two polynomials are 2 Introduction.
... polynomials'. An algebraic characterization of those systems of polynomial equations with a finite number of solutions in an algebraically closed field follows from Hilbert's Nullstellensatz: a system of polynomial equations has a ...
... polynomial in the degree and exponential in the number of variables. More ... polynomial in the number of polynomials and exponential in the dimension of the ... systems of equations and inequalities, computing the projection of a ...
Indhold
1 | |
11 | |
29 | |
SemiAlgebraic Sets | 83 |
4 | 100 |
Decomposition of SemiAlgebraic Sets | 159 |
6 | 195 |
Quantitative Semialgebraic Geometry | 237 |
Interval | 330 |
Existential Theory of the Reals | 505 |
Quantifier Elimination | 533 |
Computing Roadmaps and Connected Components of Alge | 563 |
Computing Roadmaps and Connected Components of Semi | 593 |
References | 635 |
132 | 641 |
Index of Notation 645 | 644 |
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