  We give a new method to attempt to prove 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 $n\leq 5$, thus
confirming the celebrated result of Albouy and Kaloshin.
