The classification and the computation of the zeros of quaternionic, two-sided polynomials

Abstract

Already for a long time it is known that quaternionic polynomials whose coefficients are located only at one side of the powers, may have two classes of zeros: isolated zeros and spherical zeros. Only recently a classification of the two types of zeros and a means to compute all zeros of such polynomials have been developed. In this investigation we consider quaternionic polynomials whose coefficients are located at both sides of the powers, and we show that there are three more classes of zeros defined by the rank of a certain real (4 × 4) matrix. This information can be used to find all zeros in the same class if only one zero in that class is known. The essential tool is the description of the polynomial p by a matrix equation P(z) := A(z)z + B(z), where A(z) is a real (4 × 4) matrix determined by the coefficients of the given polynomial p and P, z, B are real column vectors with four rows. This representation allows also to include two-sided polynomials which contain several terms of the same degree. We applied Newton’s method to P(z) = 0. This method turned out to be a very effective tool in finding the zeros. This method allowed also to prove, that the essential number of zeros of a quaternionic, two-sided polynomial p of degree n is, in general, not bounded by n. We conjecture that the bound is 2n. There are various examples.