We present a new approach for proving that, for a given n, there are finitely many equivalence classes of planar central configurations in the Newtonian n-body problem for generic masses. The human part of the proof relies on tropical geometry. The crux of our technique is in a computation that we have completed for n<=5, thus confirming the celebrated result of Albouy and Kaloshin.
Jensen et al. (Sat,) studied this question.