Before revision:  In exchange, the algorithm is not universally applicable : It works for polynomial system with coefficients satisfying certain concavity conditions . More precisely, it requires the given polynomial system to be located in the unbounded components of the complement of the underlying A-discriminant amoeba. A preliminary implementation of an example from the literature suggests practical performance of the algorithm. We plan to work towards a vigorous implementation including a larger scale of examples and a software paper .
After revision:  In exchange, the algorithm is not universally applicable : It works for polynomial system with coefficients satisfying certain concavity conditions .