Bounds for the zeros of quaternionic polynomials

Abstract

This paper develops new bounds for the zeros of polynomials with quaternionic coefficients. We establish eigenvalue localization theorems for quaternionic matrices and derive estimates for their spectral radii. Several explicit bounds are presented for the zeros of both simple and unilateral quaternionic polynomials, including generalizations of classical polynomial bounds. The theoretical framework employs companion matrix constructions and norm-based techniques. We demonstrate applications to stability analysis of discrete-time quaternionic systems, providing methods for estimating stability margins. Numerical examples validate the effectiveness of the proposed bounds and demonstrate their improvement over existing results in the literature.