The authors are concerned with finding bounds for the number of nonsingular solutions of a polynomial system \(f(z) = 0\) when \(f\) has a “product structure”. A theorem is proven for such systems which offers a method for obtaining a tighter upper bound on the number of nonsingular solutions than is generally available. At the same time this theorem provides an approach for solving such systems via polynomial continuation, which results in less computational work. Also, it is shown how the nine-point problem from the design of mechanisms can be solved efficiently using the present method.