Rick Moeckler gave a talk at University of Maryland on this old problem of celestial mechanics: to find the relative equilibria, ie, the configurations of N bodies of given masses which evolves under Newton law just by isometries. Lagrange was the first to find a non-trivial configuration of this type which relates to the position of the so-called Trojan asteroids.
Such configurations also provide families of periodic solutions of the N-body problem, are involves in collision asymptotics and, more deeply, they correspond to critical points of the energy-momentum map and therefore to possible topological bifurcations of the integral manifold (defined by specifying the values of the conserved quantities). It is not known whether the number of such configurations (up to a homothety) is always finite for given masses.
Smale sixth problem for the XXI century (Math. Intelligencer 1998) is precisely to prove this finiteness and compute these numbers.
Only quite special cases are known: N=1, N=2, collinear configurations (Moulton 1910, N!/2), N=4 with equal masses (Albouy 1990s), (Xia, .
THEOREM (Hampton, Moeckler) The number of planar relative equilibria of four given, arbitrary, positive masses is at least 32 and at most 8472.
It is observed that no example with more than 50 equilibria is known, even numerically.
They proved this by analyzing a system of polynomial equations (for N=4, there are six equations of Albouy-Chenciner (1988) and two extra equations of Dziobek (1900) in the six variables which describe the pairwise distances).
Classical elimination does not work (there are spurious solutions with some distances zero). They use instead the algebraic complexity theory of Bernstein, Khovansky and Kushnirenko, developped in the seventies.
If a polynomial equation has infinitely many solutions it must have a curve of solutions, described by Puiseux series. The lowest order terms of the polynomials obtained by substituting these series must then vanish. This amount to consider Minkowski sums of the Newton polytopes of the equations involved and their supporting hyperplanes. This gives rise to a large (there are 2980 facets!) but finite number of possibilities. Most of them can be discarded right away obviously no solutions). Some require the use of Gröbner basis and in a few case one has to consider higher order terms to see that there are no Puiseux series of solutions.
Finally BBK allows to bound the number of isolated solutions of a polynomila system by the so-called mixed volume of the polynomials involved.
Reference: M. Hampton, R. Moeckler: Finiteness of Relative Equilibria of the Four- body Problem, Inv. Math. 163, 289-312 (2006).