Bounds for the Zeros of Polynomials over Quaternion Division Algebra

TL;DR

This paper introduces new, sharper bounds for locating zeros of quaternionic polynomials by developing spectral norm inequalities for quaternionic matrices and applying them to companion matrices.

Contribution

It establishes novel spectral norm inequalities for quaternionic matrices and derives improved bounds for polynomial zeros, advancing quaternion polynomial analysis.

Findings

Derived sharper upper bounds for quaternionic polynomial zeros.

Established spectral norm inequalities for partitioned quaternionic matrices.

Unified framework for zero localization in quaternionic polynomials.

Abstract

Locating the zeros of quaternionic polynomials is a fundamental problem with significant implications across scientific and engineering disciplines, yet the noncommutative nature of quaternion multiplication makes it fundamentally more complex than the classical complex case. In this paper, we develop new bounds for the zeros of polynomials with quaternionic coefficients. We establish spectral norm inequalities for quaternionic matrices, particularly those of a partitioned form. These inequalities are applied to specialized quaternionic companion matrices to derive novel upper bounds for the zeros of the original polynomial. By establishing novel spectral norm inequalities for partitioned quaternionic matrices and utilizing the structural properties of companion matrices and their higher powers, we derive unexplored upper bounds for the zeros of quaternionic polynomials. Our bounds are systematically sharper than existing results and provide a unified framework for zero localization in the quaternionic setting.