Factorization of the Stability Polynomials of Ring Systems

Let $D_n$ be the dihedral group with $2n$ elements, and suppose $n$ is greater than one. We call ring system a finite $D_n$-symmetric set of points in $\mathbb{R}^2$. Ring systems have been used as models for planets surrounded by rings, and may be seen as relative equilibria of the $N$-body or the $N$-vortex problem. As a first significant step towards linear stability analysis, we study the factorization of the stability polynomial of an arbitrary ring system by systematically exploiting the ring's symmetry through representation theory of finite groups. Our results generalize contributions by J. C. Maxwell from mid-XIX century until contemporary authors such as J. Palmore and R. Moeckel, among others.