To show numerically that a generic instance of a Stewart-Gough platform has 40 isolated solutions, one can take a multihomogeneous problem formulation (described in a paper by Charles Wampler) and certify the endpoints of the solution paths defined by a multihomogeneous homotopy using alpha theory. In this talk I will describe the computation of these certificates, done jointly with Mina Khan. Exceptional situations include fewer solutions than the root count, components of solutions, and various types of singular solutions. Because the complexity of these exceptional situations may differ greatly, polyhedral methods can often give cost efficient certificates to deal with these exceptions.
FoCM 2008 -- Real Number Complexity Workshop City University of Hong Kong, 16-18 June 2008.