ELA, Volume 15, pp. 297-313, November 2006, abstract.
Zeros of Unilateral Quaternionic Polynomials
Stefano De Leo, Gisele Ducati and Vinicius Leonardi
The purpose of this paper is to show how the problem of
finding the zeros of unilateral $n$-order quaternionic
polynomials can be solved by determining the eigenvectors
of the corresponding companion matrix. This approach,
probably superfluous in the case of quadratic equations for
which a closed formula can be given, becomes truly useful
for (unilateral) n-order polynomials. To understand the
strength of this method, it is compared with the Niven
algorithm and it is shown where this (full) matrix approach
improves previous methods based on the use of the Niven
algorithm. For convenience of the readers, some examples of
second and third order unilateral quaternionic polynomials
are explicitly solved. The leading idea of the practical
solution method proposed in this work can be summarized in
the following three steps: translating the quaternionic
polynomial in the eigenvalue problem for its companion matrix,
finding its eigenvectors, and, finally, giving the quaternionic
solution of the unilateral polynomial in terms of the components
of such eigenvectors. A brief discussion on bilateral
quaternionic quadratic equations is also presented.