I've finally added a feature to Sage that I've wanted for a long time: tracking the solution paths of polynomial systems through a homotopy continuation (using Jan Verschelde's phcpack). I am cleaning up my code for formal inclusion, but it seems to work pretty well. The picture below tracks 87 of 99 solutions of the Albouy-Chenciner equations for the three-body problem (in the complex plane). The initial solutions (small blue dots) are for masses m1 = 1, m2 = 2, and m3 = 3. The final solutions are for m1 = 1/100, m2 = 1/10, and m3 = 3. Some of the solutions are moving off to infinity: the mixed volume for the system with m1=m2=0 is only 18, so 81 solutions have to coalesce or move out to infinity. (Why only 87 of the 99? The other twelve are somewhat degenerate, and their solution paths are a little jumpy). Alex Jokela helped a lot with writing the parser for the phcpack path-tracking output.