On the location of zeros of a quaternion polynomial

TL;DR

The paper addresses locating zeros of quaternionic polynomials with quaternionic coefficients. It derives Cauchy-type bounds for zeros using the maximum modulus principle and Hölder's inequality, generalizing prior complex and quaternionic results. It introduces two-parameter bounds: |q| <= (1 + (sum_{nu=0}^{n-1}|a_{nu}|^r)^{s/r})^{1/s} and |q| <= (1 + n^{s/r} M^s)^{1/s} with r>1, s>1, 1/r+1/s=1, plus special cases (e.g., r=s=2) and limits recovering known results. These bounds offer practical zero-localization for quaternionic polynomials and extend quaternion polynomial root theory to noncommutative settings.

Abstract

In this paper, we are concerned with the problem of locating the zeros of polynomials of a quaternionic variable with quaternionic coefficients. We derive some new Cauchy bounds for the zeros of a polynomial by virtue of maximum modulus theorem. Our results will generalise some recently proved results about the distribution of zeros of a quaternionic polynomial.

Theorems & Definitions (11)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Theorem 1
• Remark 1
• Corollary 1
• Theorem 2
• proof : Proof of Theorem \ref{['L2']}