Sunday, April 27, 2008
Last week I went to a very nice conference at the Bernoulli Center, on real algebraic and tropical geometry. I gave a talk on some problems on finiteness and bounds on the real solution of polynomial systems coming from the n-body problem; Sage was featured in a variety of ways during my talk. Here's an animation of a 4-body central configuration (couldn't seem to upload it to the blog directly; maybe its too big).
Thursday, April 17, 2008
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.
Saturday, April 5, 2008
Various improvements to the gfan interface in sage are in the works; one of the minor things I've had fun doing is adding more flexible color functions to the render function. Here's the Groebner fan of the 3-vortex problem relative equilibria equations, where the color is determined by the polynomial in each reduced Groebner basis which has the highest degree in any one variable - the degrees of the polynomial are converted to RGB values.