The three classes presented in this paper are by no means exhaustive, but give an idea of what can be done with homotopies to solve polynomial systems. The root counts constitute the theoretical backbone for general-purpose black-box solving. Yet, the homotopy methods are flexible enough to exploit a particular geometrical situation, with guaranteed optimal complexity when applied to generic instances.
From algebraic geometry formal procedures based on intersection theory count the number of solutions to classes of polynomial systems. Examples are the theorems of Bézout, Bernshtein and Schubert. For these situations we construct a start system and have a homotopy to deform the solutions to this start system to the solutions to any specific problem. There are many other cases for which one knows how to count but not how to deform and solve efficiently. Research in homotopy methods is aimed at turning the formal root counts into effective numerical methods. As open problem we can ask for a meta-homotopy method to connect formal root counting methods to solving generic systems and deformation procedures.
In most applications, only the real solutions are important. Once we know an optimal homotopy to solve the problem in the complex case, we would like to know whether all solutions can be real and how the real solutions are distributed. The reality question appears for instance in the theory of totally mixed Nash equilibria and in the pole placement problem. Finding well-conditioned instances of fully real problems can be done by homotopy methods. The finding of 40 real solutions to the Stewart-Gough platform [18] is perhaps the most striking example. The question is to find an efficient procedure to deform from the complex case to the fully real case.
Acknowledgments. The interest in homotopy methods by the FRISCO project has stimulated the author's research. The author is deeply indebted to all his co-authors. The interactions with Birk Huber and Frank Sottile at MSRI were influential in this current treatment of homotopies.