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Zeroes of weakly slice regular functions of several quaternionic variables on non-axially symmetric domains

Xinyuan Dou, Ming Jin, Guangbin Ren, Ting Yang

TL;DR

The paper addresses zeros of weakly slice regular functions of several quaternionic variables on non-axially symmetric domains. It develops path-slice stem functions and a $*$-product to extend holomorphy and multiplication to path spaces, and introduces path-slice conjugation and symmetrization to study zeros. Key results show that the zero set $\mathcal{Z}(f)$ of a path-slice function is contained in the zero set of its symmetrization $f^s_{\Omega_1}$, and zeros of slice regular functions form a path-slice analytic set; conjugation preserves holomorphy and slice regularity on appropriate domains. These findings extend quaternionic analysis of zeros in several variables to non-axially symmetric domains and provide structural tools for studying the analytic geometry of zero sets.

Abstract

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.

Zeroes of weakly slice regular functions of several quaternionic variables on non-axially symmetric domains

TL;DR

The paper addresses zeros of weakly slice regular functions of several quaternionic variables on non-axially symmetric domains. It develops path-slice stem functions and a -product to extend holomorphy and multiplication to path spaces, and introduces path-slice conjugation and symmetrization to study zeros. Key results show that the zero set of a path-slice function is contained in the zero set of its symmetrization , and zeros of slice regular functions form a path-slice analytic set; conjugation preserves holomorphy and slice regularity on appropriate domains. These findings extend quaternionic analysis of zeros in several variables to non-axially symmetric domains and provide structural tools for studying the analytic geometry of zero sets.

Abstract

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.
Paper Structure (5 sections, 22 theorems, 89 equations)

This paper contains 5 sections, 22 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Slice conjugation of path-slice functions
  4. Symmetrization of path-slice functions
  5. Zeros of path-slice functions

Key Result

Lemma 2.6

Dou2023003. Let $\Omega\subset\mathbb{H}_s^n$, $f\in\mathcal{PS}(\Omega)$, $\gamma\in\mathscr{P}(\mathbb{C}^n,\Omega)$, $F$ be a path-slice stem function of $f$, and $I,J\in\mathbb{S}(\Omega,\gamma)$ with $I\neq J$. Then

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 2.7
  • Definition 2.8
  • Proposition 2.9
  • Definition 2.10
  • ...and 41 more