addendum

Hi,

I have two little additions to my challenge problems:

A good reference for the equations in the Newtonian case is the paper
"Finiteness of relative equilibria of the four-body problem" by Richard
Moeckel and myself, Inventiones Mathematicae, 163, No.2, 2006, pages
289-312.

The system of 11 variables is:

G1*z1 + G2*z2 + G3*z3 + G4*z4,
k + G2*r12^2*z2 + G3*r13^2*z3 + G4*r14^2*z4,
k + G1*r12^2*z1 + G3*r23^2*z3 + G4*r24^2*z4,
k + G1*r13^2*z1 + G2*r23^2*z2 + G4*r34^2*z4,
k + G1*r14^2*z1 + G2*r24^2*z2 + G3*r34^2*z3,
1 - r12^a - r12^a*z1*z2,
1 - r13^a - r13^a*z1*z3,
1 - r14^a - r14^a*z1*z4,
1 - r23^a - r23^a*z2*z3,
1 - r24^a - r24^a*z2*z4,
1 - r34^a - r34^a*z3*z4

(all equal to zero of course).  The interesting cases are when a=2 and
a=3.  For a = 2, we are interested in solutions with positive r_{ij} and
real z_i and k, with real parameters G_i.  It would be also be interesting
to know all the solutions and their multiplicities for some parameters
(the G_i all equal for example).

I guess that's it.

Cheers,
Marshall