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 $$ α+β\cos kπ/n, \quad k=1,...,n-1, \quadβ\ne0. $$ 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 $p$: $α\pmβ\cos π/n$.