Zeros, factorizations and least common multiples of quaternion polynomials

Abstract

It is known that quaternion polynomials may have spherical zeros and isolated left and right zeros. These zeros along with appropriately defined multiplicities form the zero structure of a polynomial which can be alternatively described in terms of left and right spherical divisors of a polynomial as well as in terms of its left and right indecomposable divisors. These alternative descriptions are used to construct a polynomial with prescribed zero structure and more generally, to construct the least common multiple of given polynomials. Similar questions are discussed in the context of quaternion power series and particularly, finite Blaschke products.