Zeros of slice functions and polynomials over dual quaternions

Abstract:

This work studies the zeros of slice functions over the algebra of dual quaternions and it comprises applications to the problem of factorizing motion polynomials. The class of slice functions over a real alternative *-algebra $A$ was defined by Ghiloni and Perotti [Adv. Math. 226 (2011), pp. 1662–1691], extending the class of slice regular functions introduced by Gentili and Struppa [C. R. Math. Acad. Sci. Paris 342 (2006), pp. 741–744]. Both classes strictly include the polynomials over $A$. We focus on the case when $A$ is the algebra of dual quaternions $\mathbb {D}\mathbb {H}$. The specific properties of this algebra allow a full characterization of the zero sets, which is not available over general real alternative *-algebras. This characterization sheds some light on the study of motion polynomials over $\mathbb {D}\mathbb {H}$, introduced by Hegedüs, Schicho, and Schröcker [Mech. Mach. Theory 69 (2013), pp. 42–152] for their relevance in mechanism science.