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The classification and the computation of the zeros of quaternionic, two-sided polynomials. (English) Zbl 1190.65075
The quaternionic polynomials of the two-side type
\[ p(z):=\sum_{j=0}^n a_j z^j b_j, \quad z,a_j,b_j \in \mathbb{H}, \quad a_0b_0 \neq 0, a_n b_n \neq 0, \] where \(\mathbb{H}\) is the skew field of quaternions are treated. It is shown that there are three more classes of zeros defined by the rank of a certain real \((4\times 4)\) matrix. This information can be used to find all zeros in the same class if only one zero in that class is known. The essential tool is the description of the polynomial \(p\) by a matrix equation \(P(z):=\mathbf{A}(z)+B(z)\), where \(\mathbf{A}(z)\) is a real \((4 \times 4)\) matrix determined by the coefficients of the given polynomial \(p\) and \(P, z, B\) are real column vectors with four rows. The Newton method is applied to \(P(z)=0\). There are various examples in the paper.

65H04 Numerical computation of roots of polynomial equations
11R52 Quaternion and other division algebras: arithmetic, zeta functions
12E15 Skew fields, division rings
12Y05 Computational aspects of field theory and polynomials (MSC2010)
Full Text: DOI
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