a problem with rational functions

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).

  I will put the description of the problem as I received it at http://www.math.niu.edu/~rusin/sottocornola/ together with a Maple worksheet that constructs the six rational functions, and a flat text file which has them written out explicitly.