  We consider polynomial systems of Prony type, appearing in many areas of
mathematics. Their robust numerical solution is considered to be difficult,
especially in "near-colliding" situations. We consider a case when the
structure of the system is a-priori fixed. We transform the nonlinear part of
the Prony system into a Hankel-type polynomial system. Combining this
representation with a recently discovered "decimation" technique, we present an
algorithm which applies homotopy continuation to an appropriately chosen
Hankel-type system as above. In this way, we are able to solve for the
nonlinear variables of the original system with high accuracy when the data is
perturbed.
