  We illustrate an efficient new method for handling polynomial systems with
degenerate solution sets. In particular, a corollary of our techniques is a new
algorithm to find an isolated point in every excess component of the zero set
(over an algebraically closed field) of any $n$ by $n$ system of polynomial
equations. Since we use the sparse resultant, we thus obtain complexity bounds
(for converting any input polynomial system into a multilinear factorization
problem) which are close to cubic in the degree of the underlying variety --
significantly better than previous bounds which were pseudo-polynomial in the
classical B\'ezout bound. By carefully taking into account the underlying toric
geometry, we are also able to improve the reliability of certain sparse
resultant based algorithms for polynomial system solving.
