Numerical Algorithms in Algebraic Geometry

Polynomial systems arise in many applications: robotics, kinematics, chemical kinetics, computer vision, truss design, geometric modeling, and many others. Many polynomial systems have solutions sets, called algebraic varieties, having several irreducible components. A fundamental problem of the numerical algebraic geometry is to decompose such an algebraic variety into its irreducible components. The witness point sets are the natural numerical data structure to encode irreducible algebraic varieties. Sommese, Verschelde and Wampler represented the irreducible algebraic decomposition of an algebraic variety as a union of finite disjoint sets called numerical irreducible decomposition. The sets present the irreducible components. The numerical irreducible decomposition is implemented in Bertini . We modify this concept using partially Groebner bases, triangular sets, local dimension, and the so-called zero sum relation. We present in the second chapter the corresponding algorithms and their implementations in SINGULAR. We give some examples and timings, which show that the modified algorithms are more efficient if the number of variables is not too large

Author: Al Rashed Shwki

Publisher: Sudwestdeutscher Verlag Fur Hochschulschrifte

Publication Date: Dec 30, 2011

Number of Pages: 152 pages

Binding: Paperback or Softback

ISBN-10: 3838113500

ISBN-13: 9783838113500