A Generalization of the B\^{o}cher-Grace Theorem

TL;DR

This paper generalizes the Bôcher-Grace Theorem from third-degree polynomials to higher degrees, establishing a relationship between roots, critical points, and inscribed ellipses for convex polygons.

Contribution

It extends the classical Bôcher-Grace Theorem to polynomials of arbitrary degree with specific critical point configurations, linking roots and critical points to inscribed ellipses.

Findings

For degree n polynomials with critical points of a specific cosine form, an inscribed ellipse exists.

The ellipse's foci are the two extreme critical points of the polynomial.

The inscribed ellipse interpolates the midpoints of the convex polygon formed by the roots.

Abstract

The B\^{o}cher-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 $$ \alpha+\beta \cos k\pi/n, \quad k=1,...,n-1, \quad\beta\ne0. $$ 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 $p$: $\alpha\pm\beta \cos \pi/n$.