Bounds for the Zeros of Quaternionic Polynomials via Matrix Methods

TL;DR

This paper introduces new bounds for the zeros of quaternionic polynomials using matrix localization theorems and spectral norm techniques, providing sharper estimates and practical algorithms.

Contribution

It develops novel bounds based on localization theorems and matrix norms, improving upon classical bounds and including an algorithm with Python code for optimal zero estimation.

Findings

New bounds outperform classical estimates like Cauchy and Fujiwara

Matrix norm approach yields additional upper bounds for zeros

Algorithm predicts the sharpest bound for given polynomials

Abstract

In this paper, we derive new bounds for the zeros of quaternionic polynomials by applying localization theorems, which includes Gershgorin-type theorems for the left eigenvalues of matrices of left monic quaternionic polynomials. These results yield sharper estimates compared to existing bounds, including improvements upon Cauchy, Fujiwara and Opfer's classical bounds. Second, we develop a matrix norm approach utilizing block matrix techniques and spectral norm estimates for a specially constructed auxiliary poly nomial. This method provides additional upper bounds for polynomial zeros through careful analysis of the companion matrix's spectral radius. The comparison between the new bounds and some existing bounds have been illustrated with several examples. At the end of the paper we have given an algorithm. We have also given a Python code that predicts, for a given input which theorem will yield the sharpest upper bound. The combination of these approaches enhances the theoretical toolkit for analyzing quaternionic polynomials and offers potential applications in numerical methods, signal processing, and quaternionic quantum mechanics where zero location problems naturally arise.