Zeros and factorizations of quaternion polynomials: the algorithmic approach

Abstract: 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.