非轴对称域上多四元数变量弱切片正则函数的零点
Zeroes of weakly slice regular functions of several quaternionic variables on non-axially symmetric domains
AI总结：
本研究聚焦非轴对称域下多四元数变量弱切片正则函数的零点，提出路径切片茎函数与新型$*$-乘积，证明其零点构成路径切片解析集，推进了四元数分析相关理论。
摘要：
本研究在多四元数变量框架下研究弱切片正则函数的零点，重点关注非轴对称域。我们近期的工作提出了路径切片茎函数（path-slice stem functions）以及专为弱切片正则函数设计的新型$*$-乘积。这一创新使我们能够探索路径切片函数的共轭与对称化新技术。本研究的一项关键发现是：路径切片函数的零点完全包含在其对称化形式的零点之中，这一见解在路径切片茎函数的语境下尤为重要。我们证实，对于弱切片正则函数，一旦其切片正则性得到确认，共轭与对称化过程的作用便会凸显。此外，本研究揭示了切片正则函数零点的复杂性质：我们确定这些零点构成一个路径切片解析集，该结论源于切片正则函数的对称化的零点同样构成路径切片解析集的观测现象。这一发现推进了对四元数分析中弱切片正则函数的复杂结构与性质的理解。
英文摘要：
In this research, we study zeroes of weakly slice regular functions within the framework of several quaternionic variables, specifically focusing on non-axially symmetric domains. Our recent work introduces path-slice stem functions, along with a novel $*$-product, tailored for weakly slice regular functions. This innovation allows us to explore new techniques for conjugating and symmetrizing path-slice functions. A key finding of our study is the discovery that the zeroes of a path-slice function are comprehensively encapsulated within the zeroes of its symmetrized counterpart. This insight is particularly significant in the context of path-slice stem functions. We establish that for weakly slice regular functions, the processes of conjugation and symmetrization gain prominence once the function's slice regularity is affirmed. Furthermore, our investigation sheds light on the intricate nature of the zeroes of a slice regular function. We ascertain that these zeroes constitute a path-slice analytic set. This conclusion is drawn from the observed phenomenon that the zeroes of the symmetrization of a slice regular function also form a path-slice analytic set. This finding marks an advancement in understanding the complex structure and properties of weakly slice regular functions in quaternionic analysis.
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