Niven’s Algorithm Applied to the Roots of the Companion Polynomial Over ${{\mathbb {R}}}^4$ Algebras

Abstract

This paper will contain an extension of Niven’s algorithm of 1941, which in its original form is designed for finding zeros of unilateral polynomials p over quaternions ${\mathbb {H}}$. The extensions will cover the algebra ${{\mathbb {H}}_{\mathrm{coq}}}$ of coquaternions, the algebra ${{\mathbb {H}}_{\mathrm{nec}}}$ of nectarines and the algebra ${{\mathbb {H}}_{\mathrm{con}}}$ of conectarines. These are nondivision algebras in ${{\mathbb {R}}}^4$. In addition, it is also shown that in all algebras the most difficult part of Niven’s algorithm can easily be solved by inserting the roots of the companion polynomial c of p, with the result, that all zeros of all unilateral polynomials over all noncommutative ${{\mathbb {R}}}^4$ algebras can be found. In addition, for all four algebras the maximal number of zeros can be given. For the three nondivision algebras besides the known types of zeros: isolated, spherical, hyperbolic, a new type of zero will appear, which will be called unexpected zero of p.