Finiteness of non-degenerate central configurations of the planar $n$-body problem with a homogeneous potential

TL;DR

This paper establishes bounds on the number of non-degenerate central configurations in the planar n-body problem with a homogeneous potential, independent of the potential's degree, advancing understanding of solution multiplicity.

Contribution

It provides bounds on the count of non-degenerate central configurations in the planar n-body problem, regardless of the potential's homogeneity degree.

Findings

Existence of an upper bound for non-degenerate central configurations.

Existence of a lower bound for non-degenerate central configurations.

Bounds are independent of the potential's degree.

Abstract

We show that there exist an upper bound and a lower bound for the number of non-degenerate central configurations of the n-body problem in the plane with a homogeneous potential. In particular, both bounds are independent of the homogeneous degree of the potential under consideration.