The following problem appeared the other day on USENET. This problem comes from Nicola Sottocornola and may be described in several ways.

Algebraically, we are given 6 rational functions f_i in the function field Q(x,y,z,w) ; they must then be algebraically dependent and the question is to determine the relations among them. (In particular we wish to know whether f5 or f6 lies in the subfield generated by f1, f2, f3, and f4 .)

Geometrically, we have a rational map F : A^4 --> A^6 which should trace out an algebraic 4-fold. We want an implicitization of this variety, that is, some generators for the ideal of polynomials in 6 variables which vanishes on the 4-fold.

The applications I confess I do not really understand but they have to do with integrable systems (Henon-Heiles systems), and you can read what Nicola wrote to me in the first and second emails; these explain some of the background.

There is a Maple worksheet which constructs the six rational functions (it needs a short separate file as explained in the first email.), or you can just retrieve the six functions themselves in a flat text format. The equations include a parameter a ; I guess it would be OK to assign a random value to a if that helps make the problem solvable.

I should note that this can be written as "just" the elimination of four variables from four equations involving 10 variables, but if there is an easy way to do this, I don't know it. Even the corresponding question arising from 3 rational functions in 2 variables is too hard for a thoughtless elimination.

dave (rusin@ima.umn.edu)