Prof. Gerhard Opfer (Universität Hamburg): Zeros of unilateral quaternionic and coquaternionic polynomials

Academic or specialist Colloquium / Congress / Forum

Let ${\cal A}$ be a finite dimensional algebra over the reals.
For ${\cal A}$ we will consider
$\mathbb{H}$ (quaternions),
$\mathbb{H}_{\rm coq}$ (coquaternions),
$\mathbb{H}_{\rm nec}$ (nectarines),
and $\mathbb{H}_{\rm con}$ (conectarines),
and study the possibility of finding the zeros of unilateral polynomials
over these algebras, which are the four noncommutative algebras in~$\mathbb{R}^4$.
A polynomial $p$ will be defined by
$$p(z):=\sum_{j=0}^n a_jz^j,\quad a_j,z\in {\cal A},$$
and for finding the zeros we use of the so-called {\it companion polynomial}, which has real coefficients,
and is defined by
$$q(z):=\sum_{j,k=0}^n \overline{a_j}a_kz^{j+k}=\sum_{\ell=0}^{2n}b_\ell z^\ell \Rightarrow b_\ell\in\mathbb{R}.$$
See D. Janovsk\'a \& G. O.: SIAM J. Numer. Anal. {\bf48} (2010), 244-256,
for quaternionic polynomials and
ETNA {\bf 41} (2014), 133-158 for coquaternionic polynomials.
The real or complex roots of the companion polynomial $q$ will provide information on similarity classes which contain zeros of $p$.
It will be shown, that the companion polynomial $q$ has more capacity than formerly described in our papers, valid in all
noncommutative algebras of $\mathbb{R}^4$. There will be numerical examples.

When? 11.10.2016 17:15
Where? PER 08 Phys 2.52
Chemin du Musée 3
1700 Fribourg
Contact Department of Mathematics