  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.
