The method demonstrates finitely many equivalence classes in the Newtonian n-body problem for generic masses, indicating a novel role for tropical geometry.
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.
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Jensen et al. (2025) studied this question.
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