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Zeros of quaternion polynomials. (English) Zbl 0979.30030
For each \(q \in \mathbb H\) let \([q]\) be the equivalence class of all elements \(q'\) having the form \(xqx^{-1}\) with some \(x \in \mathbb H\). Then the authors show that for a quaternionic polynomial of the form \(f(x) = (x - x_1)\dots (x - x_n)\) the subset \(\text{zero}(f) \cap [x_r]\) is nonempty for each \(r\). This leads to a computational method to determine the zeros of the polynomial.

MSC:
30G35 Functions of hypercomplex variables and generalized variables
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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References:
[1] Zhang, F., Quaternions and matrices of quaternions, Lin. alg. and its appl., 251, 21-57, (1997) · Zbl 0873.15008
[2] Lam, T.Y., A first course in noncommutative rings, (1991), Springer Verlag, Chapter V, Section 16 · Zbl 0728.16001
[3] Niven, I., Equations in quaternions, Amer. math. monthly, 48, 654-661, (1941) · Zbl 0060.08002
[4] Serodio, R.; Pereira, E.; Vitoria, J., Computing the zeros of quaternion polynomials, () · Zbl 1050.30037
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