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.

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中文翻译：

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对偶四元数上的切片函数和多项式的零点

摘要：这项工作研究了对偶四元数代数上的切片函数的零点，它包括对分解运动多项式问题的应用。Ghiloni 和 Perotti [Adv. 数学。226 (2011), pp. 1662–1691]，扩展了 Gentili 和 Struppa [CR Math. 阿卡德。科学。巴黎 342 (2006)，第 741-744 页]。这两个类都严格包含 $A$ 上的多项式。我们关注 $A$ 是对偶四元数 $\mathbb {D}\mathbb {H}$ 的代数的情况。此代数的特定属性允许对零集进行完整表征，这在一般实替代 *-代数上是不可用的。这种表征为研究 $\mathbb {D}\mathbb {H}$ 上的运动多项式提供了一些启示，由 Hegedüs、Schicho 和 Schröcker [Mech. 马赫。Theory 69 (2013), pp. 42–152] 以表彰它们在机制科学中的相关性。

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