Finiteness of non-degenerate central configurations of the planar $n$-body problem with a homogeneous potential
Julius Natrup, Qun Wang, Yuchen Wang
TL;DR
The paper addresses the finiteness of non-degenerate central configurations in the planar $n$-body problem with homogeneous potentials $-U_{\alpha}$ for $\alpha\ge 0$. It develops an Albouy-Chenciner coordinate framework and reduces the central configuration equations to a symmetry-reduced system, then applies Khovanskii's fewnomial theory to obtain explicit uniform bounds independent of $\alpha$ and masses, namely $u(n)$ and $l(n)$. The main result is a theorem establishing both upper and lower bounds depending only on $n$, with $u(n)$ and $l(n)$ given by explicit expressions, and the work discusses optimality, small-$n$ cases, and implications for related conjectures. The findings contribute to the qualitative understanding of central configurations in generalized homogeneous $n$-body models and have potential implications for dynamics, celestial mechanics, and vortex-like systems.
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.