Coloring book reject

The image below is a Groebner fan that is just too complicated to put in the coloring book, buts I think its pretty impressive:



The ideal generating this is from what I call the "super three-vortex problem", the equations for the central configurations of a 1/r^2 potential:



I am mainly interested in the nonzero solutions of this system. To get at those, we can saturate the ideal - or in practical terms we can introduce a new variable, w, and add the equation w*s12*s13*s23 - 1 = 0 to the ideal. The 3D Groebner fan of the resulting system can be seen here.

Anders Jensen's nonregular Groebner fan in 3D

The only example of a non-regular Groebner fan that I am aware of is the following one from Anders Jensen's 2007 thesis, here plotted in 3D with Sage and gfan:

R4. = PolynomialRing(QQ,4)
idnp = R4.ideal([x*y*z+x^2*z-x*y,x*w^2-z,x*w^4+x*z])
gfnp = idnp.groebner_fan()
show(gfnp.render3d(), frame = False)

You have to follow the link for the plot since I am not sure how to include JMol applets on a blogger post.

color me gfan, now in rgb

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.

Color me Gfan

The latest version of Gfan has some new capabilities that I am excited to use for testing whether ideals are zero-dimensional. But first I have to rewrite the Sage interface to Gfan. I thought that I should try to give some Sage-added-value while I was at it, so I am converting Gfan's xfig output to Sage graphics and adding some color. Here's one result so far: a map of all the reduced Groebner bases for the 3-vortex problem, colored by maximum degree: