# scope

The workshop New Trends in Quaternions and Octonions aims to present recent advances in the research on quaternions and octonions gathering scientists working in pure as well as applied mathematics, scientific computation and applications in physics, engineering and other applied sciences.

This workshop represents an opportunity to discuss recent developments in the field of quaternions and octonions, may they be theoretical, computational or real-world applications.

The workshop will take place at University of Aveiro December 11-12 and will count with several invited lectures and short communications.

# Organization

The meeting is jointly organized by the Center for Research & Development in Mathematics and Applications (CIDMA) of University of Aveiro and the Centre of Mathematics (CMAT) of University of Minho.

## Organizers

# Invited speakers

Mouse pointer over speaker name shows title talk and abstract.

## Klaus Gürlebeck

Bauhaus-Universität Weimar, Germany
## Drahoslava Janovská

University of Chemistry and Technology, Prague, Czech Republic
## Gerhard Opfer

Department of Mathematics, University of Hamburg, Germany
## Susanne Pumplün

School of Mathematical Sciences, University of Nottingham, UK

This event is supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), within project UID/MAT/04106/2013.

## Interpolation and approximation with quaternion-valued monogenic functions

Based on abstract theory of interpolation we discuss relations between interpolation and best approximation. If the result of interpolation and best approximation coincide then we speak about optimal interpolation. Starting with this concept we study different known and some new ideas of the interpolation of monogenic functions by using Fueter polynomials, rational functions and Appell systems.

## Matrices without eigenvalues

We are focused on matrices over nondivision algebras and show by an example from an $\mathbb{R}^4$ algebra that these matrices do not necessarily have eigenvalues, even if they are invertible.
The standard condition for eigenvectors to be nonzero is replaced by the condition that $x$ contains at least one invertible component.

Our example raises several key questions:

What qualifies a matrix over a nondivision algebra to have eigenvalues?

Are these matrices diagonalizable or triangulizable? Do they allow a Schur decomposition?

## Zeros of unilateral quaternionic and coquaternionic polynomials

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 *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. Janovská and G. O.: SIAM J. Numer. Anal. **48** (2010), 244-256, for quaternionic polynomials and ETNA **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.

This research was supported by DFG, GZ OP 33/19-1.

## Factoring skew polynomials over Hamilton's quaternion algebra and the complex numbers

We show that all polynomials in a skew-polynomial ring $C[t; s, d]$ over the complex numbers decompose into a product of linear and quadratic irreducible factors. We also give a new conceptual proof for the Fundamental Theorem of Algebra for left polynomials over Hamilton's quaternion algebra.

Our proofs use nonassociative algebras constructed out of skew-polynomial rings as introduced by Petit.