A Generalization of the Bôcher-Grace Theorem

Details A Generalization of the Bôcher-Grace Theorem A Generalization of the Bôcher-Grace Theorem

2009-10-13

arxiv.org/abs/0910.2446v1

Mathematics

Complex Variables

Scientific paper

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 $p$: $\alpha\pm\beta \cos \pi/n$.

Affiliated with

Also associated with

No associations

Rating

A Generalization of the Bôcher-Grace Theorem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A Generalization of the Bôcher-Grace Theorem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A Generalization of the Bôcher-Grace Theorem will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-641421