Polynomials and Vandermonde matrices over the field of quaternions
Let \(\mathbb{H}\) denote Hamilton's quaternion division algebra over \(\mathbb{R}\). A polynomial of degree \(n\) over \(\mathbb{H}\) is a finite sum of monomials of degree \(\leq n\). By a monomial of degree \(j\) we understand a map \(m_j:\mathbb{H}\to \mathbb{H}\) given by \(m_j(x)=a_{0j}xa_{1j}xa_{2j}x\cdots a_{j-1,j}xa_{j,j}\) where \(x, a_{l,j}\in \mathbb{H}\) for \(l=0,\dots,j\). A simple quaternionic polynomial is defined as a polynomial of the type \(p_l(x)=a_0+a_1x+a_2x^2+\dots+a_n x^n\) or \(p_r(x)=a_0+xa_1+x^2a_2+\dots+x^n a_n\) where again all \(a_i,x\in \mathbb{H}\). Since the author is studying the zeros of these polynomials, he assumes w.l.o.g. that \(a_0\neq 0\) and \(a_n=1\). It is proved that in this case, the open ball \(\{z\in \mathbb{H}\,|\, |z|<r\}\) does not contain any zero of a simple polynomial \(p\), with \(r\) being the unique positive root of a certain real polynomial given by \(p\). All zeros are contained inside \(\{z\in \mathbb{H}\,|\, |z|<R\}\) with \(R=\max\{1,\sum_{j=1}^{n-1} |a_j|\}\). This generalizes results concerning the roots of complex polynomials. The second part of the paper deals with the interpolation problem and the Vandermonde matrix. For a compact set \(B\subset \mathbb{H}\) and the space \(X=C(B)\) of all quaternion valued, continuous functions defined on \(B\), it is proved that the polynomial space of all simple polynomials is not a Haar space in \(X\). This is achieved by studying two interpolation problems which can be rewritten as matrix problems using generalized Vandermonde matrices (with quaternions as entries). The author concludes the paper by pointing out that if \(V\subset X\) is a vector space of left or right dimension \(n\), then \(V\) is a Haar space if and only if all functions in \(V\setminus \{0\}\) have at most \(n-1\) zeros. This again mirrors known results for real and complex valued functions. The question whether there indeed are Haar spaces inside \(X\) remains open.
scientific article; zbMATH DE number 5692973