Zeros and factorizations of quaternion polynomials: the algorithmic approach

It is known that polynomials over quaternions may have spherical zeros and isolated left and right zeros. These zeros along with appropriately defined multiplicities form the zero structure of a polynomial. In this paper, we equivalently describe the zero structure of a polynomial in terms of its left and right spherical divisors as well as in terms of left and right indecomposable divisors. Several algorithms are proposed to find left/right zeros and left/right spherical divisors of a quaternion polynomial, to construct a polynomial with prescribed zero structure and more generally, to construct the least left/right common multiple of given polynomials. Similar questions are briefly discussed in the setting of quaternion formal power series.