On the location of zeros of a polynomial in an ellipse by Hermitian forms

TL;DR

This paper presents an algorithm to determine the number of zeros of a complex polynomial inside or outside an ellipse, generalizing earlier circle-based results.

Contribution

It introduces a method extending previous circle-based zero location results to ellipses using Hermitian forms.

Findings

Algorithm effectively counts zeros inside and outside ellipses.

Generalizes prior circle-based zero location theorems.

Applicable to complex polynomials of any degree.

Abstract

Given a complex polynomial $p(z)$ of degree $n$ and an ellipse, we find an algorithm of determining the number of zeros of $p$ in the interior and exterior of the ellipse. Our results generalize the previous results of Pt\'ak and Young about the circles.