Details:

Title	Numerical Homotopies to Compute Generic Points on Positive Dimensional Algebraic Sets
Author(s)	Andrew J. Sommese, Jan Verschelde
Type	Article in Journal
Abstract	Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to various polynomial systems, such as the cyclic n-roots problem.
Keywords	polynomial system, numerical homotopy continuation, components of solutions, numerical algebraic geometry, generic points, embedding
Length	31
Copyright	Academic Press
File
URL	dx.doi.org/10.1006/jcom.2000.0554
Language	English
Journal	Journal of Complexity
Volume	16
Number	3
Pages	572-602
Year	2000
Month	September
Translation	No
Refereed	No