A Generalization of the Bôcher-Grace Theorem

Abstract

The Bôcher-Grace Theorem can be stated as follows: Let $p$ be a third degree complex polynomial. Then there is a unique inscribed ellipse interpolating the midpoints of the triangle formed from the roots of $p$, and the foci of the ellipse are the critical points of $p$. Here, we prove the following generalization: Let $p$ be an $n^{th}$ degree complex polynomial and let its critical points take the form $$\alpha +\beta \cos k\pi /n,k=1,...,n-1,\beta \ne 0.$$ Then there is an inscribed ellipse interpolating the midpoints of the convex polygon formed by the roots of $p$, and the foci of this ellipse are the two most extreme critical points of $p$: $\alpha \pm \beta \cos \pi /n$.