## Zeros and singular points for one-sided coquaternionic polynomials with an extension to other $R^4$ algebras

Drahoslava Janovská and Gerhard Opfer

### Abstract

For finding the zeros of a coquaternionic polynomial $p$ of degree $n$, where $p$ is given in standard form $p(z)=\sum c_jz^j$, the concept of a (real) companion polynomial $q$ of degree $2n$, as introduced for quaternionic polynomials, is applied. If $z_0$ is a root of $q$, then, based on $z_0$, there is a simple formula for an element $z$ with the property that $\overline{p(z)}p(z)=0$, thus $z$ is a singular point of $p$. Under certain conditions, the same $z$ has the property that $p(z)=0$, thus $z$ is a zero of $p$. There is an algorithm for finding zeros and singular points of $p$. This algorithm will find all zeros $z$ with the property that in the equivalence class to which $z$ belongs, there are complex elements. For finding zeros which are not similar to complex numbers, Newton's method is applied, and a simple technique for computing the exact Jacobi matrix is presented. We also show, that there is no “Fundamental Theorem of Algebra” for coquaternions, but we state a conjecture that a “Weak Fundamental Theorem of Algebra” for coquaternions is valid. Several numerical examples are presented. It is also shown how to apply the given results to other algebras of $R^4$ like tessarines, cotessarines, nectarines, conectarines, tangerines, cotangerines.

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### Key words

zeros of coquaternionic polynomials, zeros of polynomials in split quaternions, companion polynomial for coquaternionic polynomials, singular points for coquaternionic polynomials, Newton method for coquaternionic polynomials, exact Jacobi matrix for coquaternionic polynomials, “Weak Fundamental Theorem of Algebra” for coquaternions, zeros of polynomials in other $R^4$ algebras (tessarines, cotessarines, nectarines, conectarines, tangerines, cotangerines)

### AMS subject classifications

12E15, 12Y05, 65J15