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

Quaternions, conceptualized by Hamilton in 1843, represent an extension of complex numbers, made possible by the Cayley-Dickson construction as noted in Dickson’s work [Dickson1919001]. This development led to a unique form of quaternionic analysis, aiming to broaden the scope of holomorphic function theory into quaternionic variable domains. One significant branch of quaternionic analysis is slice analysis over quaternions, introduced by Gentili and Struppa [Gentili2007001], which bases itself on the idea of representing the quaternionic field $\mathbb{H}$ as a collective of complex planes.

The slice regular functions, central to this theory, are functions that conform to the Cauchy-Riemann equations across these complex planes. This categorizes them as vector-valued holomorphic functions when analyzed within these planes. Notably, while simple functions like identities and polynomials qualify as slice regular functions, this classification does not extend to Fueter-regular functions, another quaternionic analysis model predating slice analysis, as discussed in Fueter’s work [Fueter1934001].

The growth and development of slice analysis have led to its integration into several mathematical disciplines, including geometric function theory [Ren2017001, Ren2017002, Wang2017001], quaternionic Schur analysis [Alpay2012001], and quaternionic operator theory [Alpay2015001, MR3887616, MR3967697, Gantner2020001, MR4496722]. Its expansion further encompasses higher dimensions through real Clifford algebras [Colombo2009002], octonions [Gentili2010001], real alternative $*$-algebras [Ghiloni2011001], and $2n$-dimensional Euclidean spaces [Dou2023002]. Additionally, the study of slice analysis extends to several variables across various frameworks [Colombo2012002, Dou2023002, Ghiloni2012001, Ghiloni2020001], presenting it as an evolution of complex analysis in several variables.

Two distinct forms of slice regular functions have emerged within this field. The first, the weakly slice regular functions, were introduced by Gentili and Struppa [Gentili2007001] and initially focused on Euclidean open sets, with the representation formula playing a pivotal role in their study [Colombo2009001]. The second, the strongly regular functions, were later introduced by Ghiloni and Perotti [Ghiloni2011001] to extend the concept to quadratic cones in real alternative $*$-algebras. This extension brought about the concept of stem functions, intrinsic to the structure of slice functions and pivotal for holomorphy and multiplication in slice analysis.

One of recent advancements in slice analysis is the slice topology [Dou2023001], a nuanced approach that transcends the limitations of the Euclidean topology and facilitates the study of slice analysis in non-axially symmetric domains. This has led to the emergence of path-slice functions and a deeper exploration of the convergence domains of quaternionic power series.

The primary objective of this paper is to investigate the zero sets of weakly slice regular functions in several quaternionic variables, particularly within non-axially symmetric slice-domains. While earlier studies [Gentili2008001, MR3026135, MR4182982] have focused on Euclidean domains, this paper aims to understand the properties of zeros on domains in slice topology. Central to this study is the concept of symmetrization of path-slice and subsequently slice regular functions. By examining the conjugation and symmetrization processes [Colombo2009001], and ensuring the preservation of slice regularity and slice-preserving properties, this paper aims to provide a thorough understanding of the zeros of these functions and their analytical nature in quaternionic variable domains.

This paper unfolds as follows: Section 2 serves as a foundation, where we delve into the fundamentals of weak slice regular functions and their associated stem functions within the context of multiple quaternionic variables. In Section 3, we introduce the concept of $\Omega_{1}$-slice conjugation for path-slice functions. We demonstrate that this conjugation maintains slice regularity when derived from a slice regular function. Section 4 is dedicated to the exploration of symmetrization within the realm of path-slice functions. Here, we establish that this symmetrization retains the critical attribute of being slice-preserving. Finally, Section 5 focuses on the zeros of a path-slice function. By juxtaposing these zeros with those of their symmetrization, we reveal that the zero set of a slice regular function forms an analytic set, thereby providing a deeper understanding of its structural properties.

This section lays the groundwork for our analysis, building upon foundational concepts and results as detailed in [Dou2023003, Dou2023004].

Denote by $\mathbb{H}$ the algebra of quaternions. Define

where

The slice topology is defined by

where $\tau(\mathbb{C}_{I}^{n})$ is the Euclidean topology in $n$-dimensional complex plane $\mathbb{C}_{I}^{n}$ and

Open sets, connected sets, and paths in $\tau_{s}$ are called respectively slice-open sets, slice-connected sets, and slice-paths.

Extending the classical complex space $\mathbb{C}^{n}$ with quaternionic imaginary units $I$ from $\mathbb{S}$ leads to the space $\mathbb{C}_{I}^{n}$. The transition employs the mapping $\Psi_{i}^{I}:\mathbb{C}^{n}\xlongrightarrow{}\mathbb{C}_{I}^{n}$, defined by

for real vectors $x,y\in\mathbb{R}^{n}$, replacing the standard imaginary unit $i$ with $I$.

To understand the structure of $\mathbb{H}_{s}^{n}$ and the behavior of related functions, we define a set of paths $\mathscr{P}(\mathbb{C}^{n})$ as continuous paths $\gamma$ from $[0,1]$ to $\mathbb{C}^{n}$, starting in the real subspace $\mathbb{R}^{n}$ at $\gamma(0)$. For a subset $\Omega$ of $\mathbb{H}_{s}^{n}$, we define $\mathscr{P}(\mathbb{C}^{n},\Omega)$ as the paths $\delta$ in $\mathscr{P}(\mathbb{C}^{n})$ for which there exists $I\in\mathbb{S}$ such that the path in $\mathbb{H}_{s}^{n}$

lies in $\Omega$.

Furthermore, for a fixed path $\gamma$ in $\mathscr{P}(\mathbb{C}^{n})$, we define the set $\mathbb{S}(\Omega,\gamma)$ as

This set contains all quaternionic imaginary units $I$ from $\mathbb{S}$ for which the path $\gamma$, when transformed to the slice $\mathbb{C}_{I}^{n}$ of $\mathbb{H}_{s}^{n}$, is entirely contained within the subset $\Omega$.

Path-slice functions and their stem functions are central to our study.

Exploring the complex structures of slice quaternionic analysis leads us to the concept of slice regularity.

By [Dou2023002, Corollary 5.9], weakly slice regular functions are path-slice.

In our exploration of specific subsets within the quaternionic cone $\mathbb{H}_{s}^{n}$, it becomes necessary to introduce and define certain key concepts.

We introduce the set $\mathscr{P}_{*}^{2}(\mathbb{C}^{n},\Omega)$, which comprises pairs of continuous paths $(\alpha,\beta)$ within $\mathscr{P}(\mathbb{C}^{n},\Omega)$ and sharing the same endpoint at $\alpha(1)=\beta(1)$. Formally, it is represented as

This function $F_{\Omega_{1}}^{f}$ is well-defined and its formulation is independent of the particular choice of the stem function $F$ within the set $\mathcal{PSS}(f)$, as established in [Dou2023003, Proposition 3.9].

For any quaternion $q=(q_{1},\ldots,q_{n})$ within the $n$-dimensional weakly slice cone $\mathbb{H}_{s}^{n}$, there exists a specific imaginary unit $I$ from the set $\mathbb{S}$ such that each component $q_{i}$ of $q$ belongs to the corresponding complex plane $\mathbb{C}_{I}$. This particular imaginary unit $I$ is denoted as $\mathfrak{I}(q)$. To define $\mathfrak{I}(q)$ precisely, consider the function:

defined by

where $\imath$ is the smallest positive integer from $\{1,\ldots,n\}$ such that the component $q_{\imath}$ of $q$ is not a real number. This formulation assigns to each quaternion in $\mathbb{H}_{s}^{n}$ an appropriate imaginary unit from $\mathbb{S}$ or the value $0$ if the quaternion lies entirely in the real space $\mathbb{R}^{n}$.

By [Dou2023003, Proposition 3.11], the following definition is well-defined.

Now we introduce the $*$-product and its properties, a fundamental operation that underpins the algebraic structure of the quaternionic function space. This operation not only enriches the algebraic framework but also provides a tool for examining the interactions between quaternionic functions in more complex scenarios.

In the context of quaternionic analysis, the standard product in $\mathbb{H}\otimes_{\mathbb{R}}\mathbb{C}$ can be extended to a unique product within the space of $2\times 1$ quaternionic column vectors, $\mathbb{H}^{2\times 1}$. This specialized product is referred to as the $*$-product. To define this product more concretely, consider two vectors $p=(p_{1},p_{2})^{T}$ and $q=(q_{1},q_{2})^{T}$ in $\mathbb{H}^{2\times 1}$. The pointwise $*$-product of these vectors is given by the formula:

where the matrices $\mathbb{I}$ and $\sigma$, and the vector $e_{1}$, are defined as

Extending this concept to functional spaces, let $\Omega$ be a subset of $\mathbb{H}_{s}^{n}$, and consider two functions $F,G:\mathscr{P}(\mathbb{C}^{n},\Omega)\rightarrow\mathbb{H}^{2\times 1}$. The functional $*$-product of $F$ and $G$ is then defined as a mapping from the set of paths $\mathscr{P}(\mathbb{C}^{n},\Omega)$ to $\mathbb{H}^{2\times 1}$, where each path $\gamma$ is mapped to the pointwise $*$-product of $F(\gamma)$ and $G(\gamma)$:

We will illustrate that within non-axially symmetric domains, the roles typically filled by axially symmetric sets can be successfully taken over by those classified as self-stem-preserving domains.

Consider complex numbers $z,w\in\mathbb{C}$ and denote $\mathcal{L}_{z}^{w}$ as the linear segment connecting $z$ to $w$. Given a path $\gamma$ in the set $\mathscr{P}(\mathbb{C}^{n})$ and a positive radius $r$, we define the set $B_{\mathscr{P}(\mathbb{C}^{n})}(\gamma,r)$ as follows:

Here, $B_{\mathbb{C}}(\gamma(1),r)$ represents the ball in the complex plane $\mathbb{C}$ centered at $\gamma(1)$ with a radius of $r$. This set essentially forms a collection of paths derived from $\gamma$ by extending its endpoint $\gamma(1)$ along all possible directions within the radius $r$ in the complex plane. For simplicity, we make the convention: when $r=0$, we set $B_{\mathscr{P}(\mathbb{C}^{n})}(\gamma,r)$ as the empty set $\phi$.

Define for $\gamma\in\mathscr{P}(\mathbb{C}^{n})$ and $r>0$,

In our study, we introduce several constants that are crucial for understanding the behavior of functions within quaternionic spaces. Let’s consider a subset $\Omega$ of the $n$-dimensional weakly slice cone $\mathbb{H}_{s}^{n}$, a path $\gamma$ in $\mathscr{P}(\mathbb{C}^{n})$, and a subset $\mathbb{S}^{\prime}$ of the set $\mathbb{S}(\Omega,\gamma)$, which includes specific quaternionic imaginary units. We define the following constants:

• •

  $r_{\gamma,\Omega}$ is defined as the supremum of all radii $r$ such that the set of paths $B_{\mathscr{P}(\mathbb{C}^{n})}(\gamma,r)$ is entirely contained within $\mathscr{P}(\mathbb{C}^{n},\Omega)$:

  $$r_{\gamma,\Omega}:=\sup\left\{r\in[0,\infty):B_{\mathscr{P}(\mathbb{C}^{n})}(\gamma,r)\subset\mathscr{P}(\mathbb{C}^{n},\Omega)\right\}.$$

• •

  $r_{\gamma,\Omega}^{\mathbb{S}^{\prime}}$ is defined as the supremum of all radii $r$ such that, for every imaginary unit $I$ in $\mathbb{S}^{\prime}$, the ball $B_{I}(\gamma^{I}(1),r)$ is contained within the slice $\Omega_{I}$:

  $$r_{\gamma,\Omega}^{\mathbb{S}^{\prime}}:=\sup\left\{r\in[0,\infty):B_{I}(\gamma^{I}(1),r)\subset\Omega_{I},\forall I\in\mathbb{S}^{\prime}\right\}.$$

• •

  $r_{\gamma,\Omega}^{2}$ is defined as the supremum among all $r_{\gamma,\Omega}^{\mathbb{S}^{\prime\prime}}$, where $\mathbb{S}^{\prime\prime}$ is a subset of $\mathbb{S}(\Omega,\gamma)$ containing at least two elements:

  $$r_{\gamma,\Omega}^{2}:=\sup\left\{r_{\gamma,\Omega}^{\mathbb{S}^{\prime\prime}}:\mathbb{S}^{\prime\prime}\subset\mathbb{S}(\Omega,\gamma)\text{ with }|\mathbb{S}^{\prime\prime}|\geqslant 2\right\}.$$

In these definitions, $B_{I}(\gamma^{I}(1),r)$ represents the set of all points $q$ in the complex plane $\mathbb{C}_{I}$ that are within a distance $r$ from the point $\gamma^{I}(1)$:

These constants play a pivotal role in determining the extent to which paths in complex space can be extended while remaining within the quaternionic subset $\Omega$.

We are expanding the idea of holomorphy of stem functions defined in complex plane $\mathbb{C}^{n}$ to encompass the one in path spaces.

We revisit some key properties of slice regular functions.

In this section, we delve into the intriguing concept of $\Omega_{1}$-slice conjugation for path-slice functions. This exploration is driven by the quest to identify an effective form of ‘conjugation’ for a given slice regular function within the confines of specific domain restrictions. While conventional notions of conjugation may not always be applicable, we discover that under certain conditions, a unique slice regular function can embody this role of conjugation. This function, termed the path-slice conjugation of the original function, emerges as a pivotal element in our analysis. Furthermore, we extend this concept to encompass path-slice functions as well, broadening the scope of our study.

This innovative approach to conjugation of slice regular functions not only enhances our understanding of their structure but also paves the way for further explorations into the properties and applications of these functions within slice quaternionic analysis.

To show that conjugation maintains the holomorphic nature of functions on path spaces, it is necessary to refer to a few important lemmas.

Let $I\in\mathbb{S}$. Then

We are now prepared to demonstrate how conjugation retains the holomorphic characteristics of functions within path spaces.