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DTSTART;VALUE=DATE:20161011T171500
DTEND;VALUE=DATE:20161011T171500
UID:5751@agenda.unifr.ch
DESCRIPTION:Let ${\cal A}$ be a finite dimensional algebra over the reals.\nFor  ${\cal A}$ we will consider \n$\mathbb{H}$ (quaternions), \n$\mathbb{H}_{\rm coq}$ (coquaternions),\n$\mathbb{H}_{\rm nec}$ (nectarines), \nand $\mathbb{H}_{\rm con}$ (conectarines),\nand study the possibility of finding the zeros of unilateral polynomials\nover these algebras, which are the four noncommutative algebras in~$\mathbb{R}^4$.\nA polynomial $p$ will be defined by\n$$p(z):=\sum_{j=0}^n a_jz^j,\quad a_j,z\in {\cal A},$$\nand for finding the zeros we use of the so-called {\it companion polynomial}, which has real coefficients,\nand is defined by \n$$q(z):=\sum_{j,k=0}^n \overline{a_j}a_kz^{j+k}=\sum_{\ell=0}^{2n}b_\ell z^\ell \Rightarrow b_\ell\in\mathbb{R}.$$\nSee D. Janovsk\'a \& G. O.: SIAM J. Numer. Anal. {\bf48} (2010), 244-256,\nfor quaternionic polynomials and\nETNA {\bf 41} (2014), 133-158 for coquaternionic polynomials. \nThe real or complex roots of the companion polynomial $q$ will provide information on similarity classes which contain zeros of $p$.\nIt will be shown, that the companion polynomial $q$ has more capacity than formerly described in our papers, valid in all\nnoncommutative algebras of $\mathbb{R}^4$. There will be  numerical examples.
SUMMARY:Prof. Gerhard Opfer (Universität Hamburg): Zeros of unilateral quaternionic and coquaternionic polynomials
CATEGORIES:Colloque / Congrès / Forum
LOCATION:PER 08\, Phys 2.52\, Chemin du Musée 3\, 1700 Fribourg
URL;VALUE=URI:https://agenda.unifr.ch/e/fr/5751
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