Smale's 6th problem for generic masses

TL;DR

This paper introduces a novel approach using tropical geometry to analyze the finiteness of equivalence classes of planar central configurations in the Newtonian n-body problem for generic masses, confirming known results for n ≤ 5.

Contribution

It presents a new method leveraging tropical geometry to prove finiteness of configurations, extending previous results and providing a computational framework.

Findings

Confirmed finiteness of configurations for n ≤ 5

Developed a tropical geometry-based computational technique

Provided a new proof approach for Smale's 6th problem

Abstract

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.