On the finiteness of four-body central configurations

TL;DR

This paper proves that the number of four-body central configurations is finite for any four nonzero masses, extending previous results that were limited to positive masses.

Contribution

It generalizes the finiteness result of four-body central configurations to include all nonzero masses, not just positive ones.

Findings

Finiteness of four-body central configurations for any nonzero masses.

Extension of previous positive-mass results to all nonzero masses.

Supports the broader understanding of four-body problem configurations.

Abstract

The number of central configurations in the four body problem was proved to be finite, first by Hampton and Moeckel, then by Albouy and Kaloshin, when the masses are all positive. We prove that the four-body central configurations are finite for any four nonzero masses.