发表机构
Università degli Studi di Bari Aldo Moro; Università di Ferrara（巴里阿尔多·莫罗大学; 费拉拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究实除代数上切片正则函数在等价类中的规范化表示，提出半模型概念，证明每个非退化函数均存在半模型，并阐明相关不变量与等价关系的作用。
AI 中文摘要
我们从自同构与共轭作用下的等价性角度，研究实除代数 $\mathbb{H}$ 和 $\mathbb{O}$ 上的切片正则函数。受近期基于不变量 $(Tr,N,cdiv)$ 的轨道理论描述，以及四元数理论中关于半正则函数的 $*$-共轭的启发，我们探讨切片正则函数是否可以被其等价类中的典范代表元所替代。本文所考虑的规范化是代数性的：我们寻找其孤立的非实零点对齐于单一复切片中的代表元。当这样的代表元属于原函数的强等价类时，我们称其为模型。我们证明纯向量函数总是存在模型，而一般而言强等价性过于刚性，此类代表元不一定存在。因此，我们引入半模型，它保持对称化以及零点集的实部和球面部分，但允许离开原强等价类。我们的主要结果证明，每个满足 $N(f)\not\equiv 0$ 的切片正则函数 $f$ 都存在半模型。该构造阐明了不变量 $(Tr,N,cdiv)$ 的不同作用，以及强等价、弱等价与半正则共轭之间的区别。
英文摘要
We study slice regular functions on the real division algebras $\mathbb{H}$ and $\mathbb{O}$ from the point of view of equivalence under automorphism and conjugation actions. Motivated by recent orbit-theoretic descriptions in terms of the invariants $(Tr,N,cdiv)$, and by the quaternionic theory of $*$-conjugation by semiregular functions, we investigate whether a slice regular function can be replaced by a canonical representative in its equivalence class. The normalization considered in this paper is algebraic: we look for representatives whose isolated nonreal zeros are aligned in a single complex slice. We call such a representative a model when it belongs to the strong-equivalence class of the original function. We prove that purely vectorial functions always admit models, while in general strong equivalence is too rigid and such representatives need not exist. We therefore introduce semi-models, which preserve the symmetrization and the real and spherical part of the zero set, but are allowed to leave the original strong-equivalence class. Our main result proves that every slice regular function $f$ with $N(f)\not\equiv 0$ admits a semi-model. The construction clarifies the different roles of the invariants $(Tr,N,cdiv)$ and the distinction between strong equivalence, weak equivalence, and semiregular conjugacy.
Comments36 pages
