The number of zeros of unilateral polynomials over coquaternions and related algebras

Janovská, Drahoslava ; Opfer, Gerhard

Electronic transactions on numerical analysis, Tome 46 (2017), pp. 55-70.

Résumé

Summary: We have proved that unilateral polynomials over the nondivision algebras in $\mathbb{R}^4$ have at most $n(2n-1)$ zeros, when the polynomial has degree $n$. Moreover, we have created an algorithm for finding all zeros of polynomials over these algebras using a real polynomial of degree $2n$, called companion polynomial. The algebras in question are coquaternions, $\mathbb{H}_{\rm coq}$, nectarines, $\mathbb{H}_{\rm nec}$, and conectarines, $\mathbb{H}_{\rm con}$. Besides the isolated and hyperbolic zeros we introduce a new type of zeros, the $unexpected$ zeros. There is a formal algorithm, and there are numerical examples. In a tutorial section on similarity we show how to find the similarity transformation of two algebra elements to be known as similar, where a singular value decomposition of a certain real $4\times4$ matrix related to the two similar elements has to be applied. We show that there is a strong indication that an algorithm by Serôdio, Pereira, and Vitória [Comput. Math. Appl., 42 (2001), pp. 1229 -- 1237], designed for finding zeros of quaternionic polynomials, is also valid in the nondivision algebras in $\mathbb{R}^4$ and it produces -- -though with another technique -- -the same zeros as those proposed in this paper.

Détail

Classification : 12D10, 12E10, 15A66
Keywords: number of zeros of polynomials over nondivision algebras in $\mathbb{R}^4$, number of zeros of polynomials over coquaternions, number of zeros of polynomials over nectarines, number of zeros of polynomials over conectarines, unexpected zeros, computation of all zeros of polynomials over nondivision algebras in $\mathbb{R}^4$