# Zeros of slice functions and polynomials over dual quaternions

## TL;DR

This paper characterizes the zeros of slice functions over dual quaternions, providing insights into motion polynomial factorization and advancing understanding of algebraic structures relevant to mechanism science.

## Contribution

It offers a full characterization of zero sets of slice functions over dual quaternions, a specific algebra, which was previously unknown for general alternative $*$-algebras.

## Key findings

- Full zero set characterization for dual quaternion slice functions
- Enhanced understanding of motion polynomial factorization
- Implications for mechanism science applications

## 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 an alternative $*$-algebra $A$ was defined by Ghiloni and Perotti in 2011, extending the class of slice regular functions introduced by Gentili and Struppa in 2006. Both classes strictly include the polynomials over $A$. We focus on the case when $A$ is the algebra of dual quaternions $\mathbb{DH}$. The specific properties of this algebra allow a full characterization of the zero sets, which is not available over general alternative $*$-algebras. This characterization sheds some light on the study of motion polynomials over $\mathbb{DH}$, introduced by Heged\"us, Schicho, and Schr\"ocker in 2013 for their relevance in mechanism science.