A generalization of the Bocher-Grace theorem

Abstract

The Bocher-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 \[ \alpha +\beta \cos k\pi /n,\quad k=1,\ldots ,n-1,\ \beta \ne 0. \] 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 $p$: $\alpha \pm \beta \cos \pi /n$.