Annular bounds for the zeros of a polynomial from companion matrix

TL;DR

This paper introduces new upper bounds for the moduli of polynomial zeros based on the companion matrix, enabling tighter annuli that contain all zeros, improving upon existing bounds.

Contribution

The paper develops sharper upper bounds for polynomial zeros' moduli using companion matrix analysis, refining previous bounds and allowing for smaller zero-containing annuli.

Findings

New bounds are sharper than previous results in certain cases.

Bounds enable the description of smaller annuli containing all zeros.

The bounds depend on the coefficients' norms and provide improved zero localization.

Abstract

Let $p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\ldots+a_1z+a_0$ be a complex polynomial with $a_0\neq 0$ and $n\geq 3$. Several new upper bounds for the moduli of the zeros of $p$ are developed. In particular, if $\alpha=\sqrt{\sum_{j=0}^{n-1}|a_j|^2}$ and $z$ is any zero of $p$, then we show that \begin{eqnarray*} |z|^2 &\leq & \cos^2 \frac{\pi}{n+1}+|a_{n-2}|+ \frac{1}{4} \left ( |a_{n-1}|+ { \alpha} \right)^2 + \frac{1}{2}\sqrt{\alpha^2-|a_{n-1}|^2} + \frac{1}{2}{\alpha}, \end{eqnarray*} which is sharper than the Abu-Omar and Kittaneh's bound \begin{eqnarray*} |z|^2 &\leq & \cos^2 \frac{\pi}{n+1}+ \frac{1}{4} \left ( |a_{n-1}|+ { \alpha}\right)^2 + {\alpha} \end{eqnarray*} if and only if $2|a_{n-2}|< \sqrt{\sum_{j=0}^{n-1}|a_j|^2}-\sqrt{\sum_{j=0}^{n-2}|a_j|^2}. $ The upper bounds obtained here enable us to describe smaller annuli in the complex plane containing all the zeros of $p$.