Classifying zeros of two-sided quaternionic polynomials and computing zeros of two-sided polynomials with complex coefficients

Abstract

We improve the method of Janovska and Opfer for computing the zeros on the surface of a given sphere for a quaternionic two-sided polynomial. We classify the zeros of quaternionic two-sided polynomials into three types — isolated, spherical and circular — and characterize each type. We provide a method to find all quaternion zeros for two-sided polynomials with complex coefficients. We also establish standard formulae for roots of a quadratic two-sided polynomial with complex coefficients, which yields a simpler and more efficient algorithm to produce all zeros in the quadratic case.