A homotopy is a family of polynomial systems which defines a deformation from a system with known solutions to a system whose solutions are needed. Via dynamic load balancing we may distribute the solution paths so that a close to optimal speed up is achieved. Polynomial systems -- such as the 9-point problem in mechanical design leading to 286,720 paths -- whose solving required real supercomputers twenty years ago can now be handled by modest personal cluster computers, and soon by multicore multiprocessor workstations. Larger polynomial systems however may lead to more numerical difficulties which may skew the timing results, so that attention must be given to "quality up" as well. Modern homotopy methods consist of sequences of different families of polynomial systems so that not only the solution paths but also parametric polynomial systems must be exchanged frequently.
Interactive Parallel Computation in Support of Research in Algebra, Geometry and Number Theory. MSRI, 29 January - 2 February 2007