## Polynomials and Vandermonde matrices over the field of quaternions

Gerhard Opfer

### Abstract

It is known that the space of real valued, continuous functions $C(B)$ over a multidimensional compact domain $B\subset{\bf R}^k\>,k\ge2$ does not admit Haar spaces, which means that interpolation problems in finite dimensional subspaces $V$ of $C(B)$ may not have a solutions in $C(B)$. The corresponding standard short and elegant proof does not apply to complex valued functions over $B\subset{\bf C}$. Nevertheless, in this situation Haar spaces $V\subset C(B)$ exist. We are concerned here with the case of quaternionic valued, continuous functions $C(B)$ where $B\subset{\bf H}$ and ${\bf V}$ denotes the skew field of quaternions. Again, the proof is not applicable. However, we show that the interpolation problem is not unisolvent, by constructing quaternionic entries for a Vandermonde matrix ${\bf V}$ such that ${\bf V}$ will be singular for all orders $n>2$. In addition, there is a section on the exclusion and inclusion of all zeros in certain balls in ${\bf H}$ for general quaternionic polynomials.

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### Key words

Quaternionic interpolation polynomials, Vandermonde matrix in quaternions, location of zeros of quaternionic polynomials

### AMS subject classifications

11R52, 12E15, 12Y05, 65D05

### Links to the cited ETNA articles

 [8] Vol. 26 (2007), pp. 82-102 Drahoslava Janovská and Gerhard Opfer: Computing quaternionic roots by Newton's method