(Semi-)Models for Slice Regular Functions on Real Division Algebras
Abstract
We study slice regular functions on the real division algebras ℍ and 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≡ 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.
Related
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- Two-sided slice regular functions
- A connection between quaternionic slice regular and Fueter hyperholormorphic functions
- A four-dimensional residue calculus for quaternionic slice regular functions
- Classifying Slice-Regular Polynomials via Group Actions on the Twistor Space
BibTeX
@misc{altavilla2026semimodels,
title = {{(Semi-)Models for Slice Regular Functions on Real Division Algebras}},
author = {Amedeo Altavilla and Cinzia Bisi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06463},
eprint = {2609.06463},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06463},
}
RIS
TY - UNPB
TI - (Semi-)Models for Slice Regular Functions on Real Division Algebras
AU - Amedeo Altavilla
AU - Cinzia Bisi
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06463
UR - https://arxiv.org/abs/2609.06463
AB - We study slice regular functions on the real division algebras ℍ and 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≡ 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.
ER -