Smale's 6th Problem for Generic Masses

Abstract

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.