# Vanishing of quaternionic cohomology groups and applications

Jasna Prezelj  UL FMF, Jadranska 19, Ljubljana, Slovenija, UP FAMNIT,
Glagoljaška 8, Koper, Slovenija, IMFM, Jadranska 19, Ljubljana, Slovenija
<EMAIL_ADDRESS>and  Fabio Vlacci  MIGe Università di Trieste
Piazzale Europa 1, Trieste, Italy<EMAIL_ADDRESS>
###### Abstract.

We present solutions to additive and multiplicative Cousin problems formulated
on an axially symmetric domain $\Omega\subset\mathbb{H}$ for slice–regular
functions starting from the solutions for subclasses, namely slice–regular
slice–preserving functions and functions in a given vectorial class. As a
consequence, we prove the vanishing of the corresponding cohomology groups
with respect to axially symmetric open coverings (Theorems 1.1, 4.1, 4.2). The
main tool used in the proofs of these theorems is the existence of
quaternionic Cartan coverings and Cartan’s splitting lemmas. As an application
we prove a jet interpolation theorem (Theorem 1.2) and we show that every
divisor is principal (Theorem 5.1).

###### Key words and phrases:

Cartan coverings, Cousin problems, cohomology groups, divisors

###### 1991 Mathematics Subject Classification:

30G35, 14C20

The first author was partially supported by research program P1-0291 and by
research projects N1-0237, J1-3005 at Slovenian Research Agency. The second
author was partially supported by Progetto MIUR di Rilevante Interesse
Nazionale PRIN 2010-11 Varietà reali e complesse: geometria, topologia e
analisi armonica. The research that led to the present paper was partially
supported by a grant of the group GNSAGA of Istituto Nazionale di Alta
Matematica ‘F: Severi’.

##  1\. Introduction

The theory of slice–regular functions (denoted by $\mathcal{SR}$ and shortly
recalled in the Section 2 and mainly focussed on specific aspects useful
later) clearly shows many promising aspects to become a good framework to
consider the generalizations of Cousin problems for quaternionic functions;
indeed, the notion of slice–regularity is a good extension of the notion of
holomorphicity for quaternionic functions and semi-regular functions play the
role of meromorphic functions. The analogue of Weierstrass and Mittag-Leffler
Theorems are already at our disposal (see [GSS]). Furthermore, domains of
holomorphicity are generalized in terms of quaternionic axially symmetric
domains.

In the present paper we prove the analogues of the above theorems for
arbitrary axially symmetric domains $\Omega\subset\mathbb{H}.$

With the proper definition of sheaves (Section 3) and cohomology, namely, the
cohomology with respect to axially symmetric coverings (Definition 2.2), we
can formulate the first main theorem of this paper:

######  Theorem 1.1 (Main Theorem 1).

Let $\Omega$ be an axially symmetric domain. Then

$H^{1}(\Omega,\mathcal{SR}^{*})=0.$

In quaternionic settings we have two types of jets, usual ones at a point,
i.e. the value and some derivatives at a point are given, and spherical jets,
given by finite spherical expansions (see [GSS]). Theorem 1.1 yields the
second main theorem of this paper,

######  Theorem 1.2 (Main Theorem 2).

Let $\Omega$ be an axially symmetric domain and $Z$ a discrete set of points
and spheres. Then there exists $f\in\mathcal{SR}(\Omega)$ with prescribed
finite jets at $Z.$

In particular, on any axially symmetric domain, there exists functions with
prescribed zeroes (Theorem 5.2). Another consequence of Theorem 1.1 is the
existence of principal divisors, i.e. functions with prescribed zeroes and
poles (Theorem 5.1).

The main method applied in the proofs of these theorems is gluing local
solutions. To this end we prove additive and multiplicative Cartan’s Splitting
lemmas for various classes of slice–regular functions and use them for solving
first and second Cousin problems for the aforementioned classes of
slice–regular functions (Section 4, Theorems 4.2, 4.1). To deal with the
Cousin problems one has to work with Cartan coverings (Section 4 in [PV2]). In
the last section, Section 5, we apply our results to prove theorems, which are
analogous to results on divisors and jet interpolation theorems on Stein
manifolds.

##  2\. Preliminary results

We denote by $\mathbb{H}$ the algebra of quaternions. Let $\mathbb{S}$ be the
sphere of imaginary units, i.e. the set of quaternions $I$ such that
$I^{2}=-1$. Given any quaternion $z\not\in\mathbb{R},$ there exist (and are
uniquely determined) an imaginary unit $I$, and two real numbers $x,y$ (with
$y\geq 0$) such that $z=x+Iy$. With this notation, the conjugate of $z$ will
be $\bar{z}:=x-Iy$ and $|z|^{2}=z\bar{z}=\bar{z}z=x^{2}+y^{2}$. Each imaginary
unit $I$ generates (as a real algebra) a copy of a complex plane denoted by
$\mathbb{C}_{I}:=\mathbb{R}+I\mathbb{R}$. We call such a complex plane a
slice. The upper half-plane in $\mathbb{C}_{I}$, namely $\\{x+yI\ :\ y>0\\}$
will be denoted by $\mathbb{C}_{I}^{+}$ and called a leaf. Set
$\mathbb{C}_{I}^{-}:=\mathbb{C}_{-I}^{+}$ and for a subset
$E\subset\mathbb{H}$ define $E_{I}:=E\cap\mathbb{C}_{I},$
$E_{I}^{+}:=E\cap\mathbb{C}_{I}^{+},$ $E_{I}^{-}:=E\cap\mathbb{C}_{I}^{-},$
$E_{I}^{0}:=E\cap\mathbb{R}.$ We set
$\mathbb{S}(a,r):=\\{a+rI,I\in\mathbb{S}\\}=a+r\mathbb{S}$ for
$a,r\in\mathbb{R}$. The _(axial) symmetrization_ $\widetilde{E}$ of a subset
$E$ of $\mathbb{H}$ is defined by

$\widetilde{E}=\\{x+Iy:x,y\in\mathbb{R},I\in\mathbb{S},(x+\mathbb{S}y)\cap
E\neq\varnothing\\}.$

A subset $\Omega$ of $\mathbb{H}$ is called (axially) symmetric (in
$\mathbb{H}$) if $\widetilde{\Omega}=\Omega.$ A subset $U$ of $\mathbb{C}$ is
called symmetric if it is invariant for the reflection $R$ over the real axis,
i.e. $U=R(U)$.

In the sequel we briefly recall some standard definitions (see [GSS], [GPV]).

Natural domains of definition for slice–regular functions are slice domains. A
domain $\Omega$ of $\mathbb{H}$ is called a _slice domain_ if, for all
$I\in\mathbb{S}$, the subset $\Omega_{I}$ is a domain in $\mathbb{C}_{I}$ and
$\Omega^{0}=\Omega\cap\mathbb{R}\neq\varnothing$. If, moreover, $\Omega$ is
axially symmetric, then it is called a symmetric slice domain. Slice functions
(see [GP]) are naturally defined also on product domains, i.e. axially
symmetric domains of $\mathbb{H}\setminus\mathbb{R}$. Hence an axially
symmetric domain $\Omega$ is either a symmetric slice domain or it is a
product domain.

The present paper focuses on axially symmetric domains only and slice–regular
functions defined on such sets. Contrary to the complex case, where a ball is
a natural local domain of definition for holomorphic functions, in the case of
slice–regular functions the natural domains, called basic domains ([GPV]) are
symmetrizations of topological discs in $\mathbb{C}_{I}\setminus\mathbb{R}$
(basic product domains) or symmetrizations of symmetric topological discs in
$\mathbb{C}_{I}$ (basic slice domains), $I\in\mathbb{S}$. A basic domain is
also a _basic neighborhood_ of any of its points. We define also the empty set
to be a basic set. An axially symmetric closed set $V$ of $\mathbb{H}$ is
called a closed basic set if it is either a symmetrization of a closed
topological discs in $\mathbb{C}_{I}\setminus\mathbb{R}$ or a symmetrization
of a symmetric closed topological disc in $\mathbb{C}_{I}$ for some
$I\in\mathbb{S}$.

Notice that the intersection of a basic domain with the real axis is either
empty or connected. The closure of a basic set is not necessarily a closed
basic set. A closed basic set with a piecewise smooth boundary has a basis of
basic sets.

######  Definition 2.1 (Symmetric open covering).

Let $U\subset\mathbb{C}$ be a symmetric domain and let
$\mathcal{U}=\\{U_{\lambda}\\}_{\lambda\in\Lambda}$ be an open covering of
$U.$ The covering $\mathcal{U}$ is a symmetric open covering if each
$U_{\lambda}\in{\mathcal{U}}$ is a symmetric open set.

######  Definition 2.2 (Axially symmetric open covering).

Let $\Omega\subset\mathbb{H}$ be an axially symmetric domain and let
$\mathcal{U}=\\{U_{\lambda}\\}_{\lambda\in\Lambda}$ be an open covering of
$\Omega.$ The covering $\mathcal{U}$ is an axially symmetric open covering if
each $U_{\lambda}\in{\mathcal{U}}$ is an axially symmetric open set.

If $\iota^{2}=-1,$ consider the complexification
$\mathbb{H}_{\mathbb{C}}=\mathbb{H}+\iota\mathbb{H},$ of the skew field
$\mathbb{H}$ and set $\zeta\colon z=x+\iota y\mapsto x-\iota y=:\bar{z}$ to be
the natural involution of $\mathbb{H}_{\mathbb{C}}$. For any $J\in\mathbb{S}$,
define the map

$\phi_{J}\colon\mathbb{H}_{\mathbb{C}}\to\mathbb{H},\quad\phi_{J}(x+\iota
y)=x+Jy$

Notice that the map $\phi_{J}$, when restricted to
$\mathbb{R}_{\mathbb{C}}\cong\mathbb{R}+\iota\mathbb{R}\cong\mathbb{C}$, is an
isomorphism between $\mathbb{C}$ and $\mathbb{R}+J\mathbb{R}=\mathbb{C}_{J}$.

If $\Omega\subseteq\mathbb{H}$ is an axially symmetric domain, and if $i$
denotes the imaginary unit of $\mathbb{C}\subset\mathbb{H}$, then the
intersection
$\Omega_{i}=\Omega\cap(\mathbb{R}+i\mathbb{R})=\Omega\cap\mathbb{C}$ defines a
domain of the complex plane that is invariant under complex conjugation. With
respect to the established notations, the subset $\Omega_{\iota}=\\{x+\iota
y\in\mathbb{H}+\iota\mathbb{H}:x+iy\in\Omega_{i}\\}$ is called the _image of
$\Omega_{i}$ in $\mathbb{H}_{\mathbb{C}}$_, and is invariant under involution,
i.e., $\Omega_{\iota}=\zeta(\Omega_{\iota})$. We are now in a position to
recall the following definitions.

######  Definition 2.3 (Stem function).

Let $\Omega\subseteq\mathbb{H}$ be an axially symmetric open set, let
$\Omega_{i}=\Omega\cap(\mathbb{R}+i\mathbb{R})$ and let $\Omega_{\iota}$ be
the image of $\Omega_{i}$ in $\mathbb{H}_{\mathbb{C}}$.

A (continuous) function $F\colon\Omega_{\iota}\to\mathbb{H}_{\mathbb{C}}$ is
called a _stem function_ if $F(\bar{z})=\overline{F(z)}$ for all
$z\in\Omega_{\iota}$. For each stem function
$F\colon\Omega_{\iota}\to\mathbb{H}_{\mathbb{C}}$, there exists a unique
$f\colon\Omega\to\mathbb{H}$ such that the diagram

${\Omega_{\iota}}$${\mathbb{H}_{\mathbb{C}}}$${\Omega}$${\mathbb{H}}$$\scriptstyle{\phi_{J}}$$\scriptstyle{F}$$\scriptstyle{\phi_{J}}$$\scriptstyle{f}$

commutes for all $J\in\mathbb{S}$. The function $f$ is called the _slice
function_ induced by $F$ and denoted by $\mathcal{J}(F)$. The set of all slice
functions on $\Omega$ is denoted by $\mathcal{S}(\Omega).$

Let $f=\mathcal{J}(F),g=\mathcal{J}(G)$ be the slice functions induced by the
stem functions $F,G$ respectively. The _$*$ -product_ of $f$ and $g$ is
defined as the slice function $f*g:=\mathcal{J}(FG)$.

We will use definitions of slice–regularity and of $*$-product that involve
stem functions, and that are valid for any axially symmetric domain of
$\mathbb{H}$. When restricted to symmetric slice domains, slice–regularity
coincides with the Definition 1.2 of slice–regularity initially presented in
[GS].

######  Definition 2.4 (Slice–regular function).

Let $\Omega\subseteq\mathbb{H}$ be an axially symmetric open set. A slice
function $f\colon\Omega\to\mathbb{H}$, induced by a stem function
$F\colon\Omega_{\iota}\to\mathbb{H}_{\mathbb{C}}$, is called _slice–regular_
if $F$ is holomorphic. The set of all slice–regular functions on $\Omega$ is
denoted by $\mathcal{SR}(\Omega).$

A slice function $f\colon\Omega\to\mathbb{H}$ is said to be slice–preserving
if and only if $\forall I\in\mathbb{S},\forall
z\in\Omega_{I}:=\Omega\cap\mathbb{C}_{I}$ we have that
$f(z)\in\mathbb{C}_{I}$. The set of all slice–regular functions, which are
slice–preserving in $\Omega,$ will be denoted as
$\mathcal{S}\mathcal{R}_{\mathbb{R}}(\Omega)$. The set
$\mathcal{SR}^{*}_{\mathbb{R}}(\Omega)\subset\mathcal{SR}_{\mathbb{R}}(\Omega)$
denotes the subset set of all nonvanishing functions and
$\mathcal{SR}^{+}_{\mathbb{R}}(\Omega)\subset\mathcal{SR}_{\mathbb{R}}(\Omega)$
denotes the set of all functions $\mathcal{SR}_{\mathbb{R}}(\Omega)$ which are
strictly positive on the real axis.

The sup-norm for a bounded function $f$ on $\Omega$ is denoted by
$|f|_{\Omega}$.

######  Definition 2.5 ([GSS], Definition 5.21).

A function $f$ is semiregular in a symmetric slice domain $\Omega$ if it is
regular in a symmetric slice domain $\Omega^{\prime}\subset\Omega$ such that
every point of $\Omega\setminus\Omega^{\prime}$ is a pole (or a removable
singularity) for $f$. We denote the set of semiregular functions on $\Omega$
by $\mathcal{SMR}(\Omega).$

######  Remark 2.6.

The (conceptual) correspondence between the classes $\mathcal{C}(\Omega),$
$\mathcal{O}(\Omega)$ and $\mathcal{M}(\Omega)$ of (complex) continuous,
holomorphic and meromorphic functions and the classes $\mathcal{S}(\Omega),$
$\mathcal{SR}(\Omega)$ and $\mathcal{SMR}(\Omega)$ of (quaternionic) slice,
slice–regular and semiregular functions on $\Omega$ is presented below.

$\mathcal{C}(\Omega)\leftrightarrow\mathcal{S}(\Omega)\qquad\mathcal{O}(\Omega)\leftrightarrow\mathcal{SR}(\Omega)\qquad\mathcal{M}(\Omega)\leftrightarrow\mathcal{SMR}(\Omega).$

As shown in [GSS], the zero set of any $f\in\mathcal{SR}(\Omega),\ f\neq 0$,
is a discrete set of points and spheres of the form $S(a,r)$ contained in
$\Omega$. At each point $q_{0}\in\Omega$, by Theorem 2.12 in [GSS], the
function $f$ has a Taylor series expansion of the form

$f(q)=\sum_{n=0}^{\infty}(q-q_{0})^{*n}f^{(n)}(q_{0})/n!,$

which converges on some $\sigma$-ball $\Sigma(q_{0},R)$. The notion of a jet
of $f$ of order $l$ at a point $q_{0}$ then consists of a sequence of numbers
$A_{0},A_{1},\ldots A_{l}$ representing the values
$f^{(n)}(q_{0})/n!=A_{n},n=0,\ldots,l$.

By Theorem 8.7 in [GSS], the function also has a spherical expansion at a
point $q_{0}\in x_{0}+y_{0}\mathbb{S}$ of the form

$f(q)=\sum_{n=0}^{\infty}[(q-x_{0})^{2}+y_{0}^{2}]^{n}[A_{2n}+(q-q_{0})A_{2n+1}],$

which converges on a Cassini ball
$U(x_{0}+y_{0}\mathbb{S},R)=\\{q\in\mathbb{H},|(q-x_{0})^{2}+y_{0}^{2}|<R^{2}\\}\subset\Omega$.
We define a spherical jet of order $2l+1$ at $q_{0}$ to be the sequence
$A_{0},\ldots,A_{2l+1}$, representing the finite part of spherical expansion.
If $A_{2m}$ or $A_{2m+1}$ are the first nonzero coefficients in the above
series expansion, then $2m$ is the spherical multiplicity of $f$ at
$x_{0}+y_{0}\mathbb{S}$.

If the point $q_{0}$ on the sphere is not given, then we define a spherical
jet of order $2l+1$ on the sphere $S(x_{0},y_{0})$ to be a sequence
$A_{0},A_{1},\ldots A_{2l},A_{2l+1}$ with $A_{k}=0$ for odd $k$.

A spherical interpolation theorem (with jets of order $0$) with approximation
for entire slice-regular functions on $\mathbb{H}$ was proved in [PV1].

The next proposition recalls two well known technical results that will be
extensively used in the sequel (see, e.g., [GP]).

######  Proposition 2.7.

Let $\Omega\subseteq\mathbb{H}$ be an axially symmetric open set, and let
$f,g\in\mathcal{SR}(\Omega)$ be two slice–regular functions. Then

  1. (a)

the $*$-product $f*g$ is a slice–regular function on $\Omega$;

  2. (b)

if $f$ is slice–preserving, then $f*g=fg=g*f$, i.e, the $*$-product coincides
with the pointwise product;

  3. (c)

if $f(q_{0})=0,$ then $(f*g)(q_{0})=0$.

The _imaginary unit function_
$\mathcal{I}\colon\mathbb{H}\setminus\mathbb{R}\to\mathbb{S}$ is defined by
setting $\mathcal{I}(q)=I$ if $q\in\mathbb{C}_{I}^{+}.$ The function
$\mathcal{I}$ is slice–regular and slice–preserving, but it is not an open
mapping and it is not defined on any slice domain.

Consider now an axially symmetric open set $\Omega$ and
$f\in\mathcal{SR}(\Omega).$ We have already defined the splitting
$f=f_{0}+f_{v},$ where the scalar part $f_{0}$ of $f$ is a slice–preserving
function.

######  Definition 2.8.

The function $f\in\mathcal{SR}(\Omega)$ is a vectorial function if $f=f_{v}.$
The set of vectorial functions on $\Omega$ will be denoted by
$\mathcal{S}\mathcal{R}^{v}(\Omega)$. We have
$\mathcal{SR}(\Omega)=\mathcal{SR}_{\mathbb{R}}(\Omega)\oplus\mathcal{SR}^{v}(\Omega).$

In [GPV] the notion of a vectorial class was introduced (compare Definitions
5.1, 5.2 and 5.3 therein). Here we extend the existing definition.

######  Definition 2.9.

Let $f_{v}\in\mathcal{S}\mathcal{R}^{v}(U)$ and
$g_{v}\in\mathcal{S}\mathcal{R}^{v}(U^{\prime}),$ where
$U,U^{\prime}\subset\mathbb{H}$ are axially symmetric domains in $\mathbb{H}$
such that $U\cap U^{\prime}\neq\varnothing$. Take $\widetilde{p}\subset U\cap
U^{\prime}$; we say that $f_{v}$ and $g_{v}$ are _equivalent at
$\widetilde{p}$_, in symbols $f_{v}\sim_{\widetilde{p}}g_{v}$, if there exist
an axially symmetric neighborhood $V_{\widetilde{p}}$ of $\widetilde{p}$,
$V_{\widetilde{p}}\subset U\cap U^{\prime}$, such that $f_{v}$ and $g_{v}$ are
linearly dependent over
$\mathcal{S}\mathcal{R}_{\mathbb{R}}(V_{\widetilde{p}})$ in
$V_{\widetilde{p}}$. We will denote by $[f_{v}]_{\widetilde{p}}$ the
$\sim_{\widetilde{p}}$ equivalence class whose representative is $f_{v}$.

Loosely speaking, a global vectorial class $\omega$ on an open axially
symmetric set $\Omega$ is determined by choosing a set of vectorial functions
on an open axially symmetric covering
$\mathcal{U}=\\{U_{\lambda}\\}_{\lambda\in\Lambda}$ of $\Omega$, namely the
set
$\mathcal{V}=\\{f_{\lambda}\in\mathcal{SR}^{v}(U_{\lambda}),U_{\lambda}\in\mathcal{U}\\}$,
so that if $f_{\lambda},f_{\mu}\in\mathcal{V}$, then $[f_{\lambda}]=[f_{\mu}]$
on $U_{\lambda}\cap U_{\mu}$. By Remark 4.2 in [GPV], for each vectorial class
$\omega$, we can choose a local representative $f_{v}$ having neither real nor
spherical zeroes. Such representative is called a minimal representative of
$\omega$. We denote
$\mathcal{SR}_{\omega}(\Omega):=\\{f\in\mathcal{SR}(\Omega),[f_{v}]=\omega\\}.$
Notice that $\mathcal{SR}_{\mathbb{R}}=\mathcal{SR}_{[0]}$.

A direct calculation shows that for $f,g\in\mathcal{SR}_{\omega}(U),$ we have
$f*g=g*f$ (see also [AdF1]).

We will prove later (Theorem 4.10), that every vectorial class on an axially
symmetric open set $\Omega$ has a global minimal representative.

######  Example 2.10.

The class of slice–regular functions on an axially symmetric domain $\Omega$
which preserve the slice $\mathbb{C}_{i}$ coincides with
$\mathcal{SR}_{[i]}(\Omega)$ because each function, which preserves the slice
$\mathbb{C}_{i}$, is of the form $f=f_{0}+f_{1}i$ and thus defines an
embedding of the set of holomorphic functions on symmetric domains into the
set of slice–regular functions on axially symmetric domains.

Given a standard basis of $\mathbb{H},$ the vectorial part of a slice–regular
function can be decomposed further ([GMP], [CGS], [AdF1]): indeed, the next
result holds for slice functions in general (see [GMP]).

######  Proposition 2.11.

Let $\\{1,i,j,k\\}$ be the standard basis of $\mathbb{H}$ and assume $\Omega$
is an axially symmetric domain of $\mathbb{H}$. Then the map

$(\mathcal{S}\mathcal{R}_{\mathbb{R}}(\Omega))^{4}\ni(f_{0},\ f_{1},\ f_{2},\
f_{3})\mapsto f_{0}+f_{1}i+f_{2}j+f_{3}k\in\mathcal{S}\mathcal{R}(\Omega)$

is bijective.

In the sequel, all bases of $\mathbb{H}\cong\mathbb{R}^{4}$ will be
orthonormal (and positively oriented) with respect to the standard scalar
product of $\mathbb{R}^{4}$. Proposition 2.11 implies that, given any
$f,g\in\mathcal{SR}(\Omega)$, there exist unique
$f_{l},g_{l}\in\mathcal{S}\mathcal{R}_{\mathbb{R}}(\Omega),l=0,\ldots,3$, such
that

$\displaystyle f=f_{0}+f_{1}i+f_{2}j+f_{3}k=f_{0}+f_{v}$ $\displaystyle
g=g_{0}+g_{1}i+g_{2}j+g_{3}k=g_{0}+g_{v}$

With the above given notation, if we call _regular conjugate_ of $f$ the
function $f^{c}=f_{0}-f_{v}$ then we have

(2.1) $f_{0}=\frac{f+f^{c}}{2}.$

Furthermore, using Definition 2.3 and Proposition 2.4, we obtain the following
expression for the $*$-product of $f$ and $g$:

(2.2)
$f*g=f_{0}g_{0}-f_{1}g_{1}-f_{2}g_{2}-f_{3}g_{3}+f_{0}g_{v}+g_{0}f_{v}+\dfrac{f_{v}*g_{v}-g_{v}*f_{v}}{2}$

or, in terms of components with respect to the basis $\\{1,i,j,k\\}$,

(2.3)
$\begin{array}[]{ccc}(f*g)_{0}&=&f_{0}g_{0}-f_{1}g_{1}-f_{2}g_{2}-f_{3}g_{3}\\\
(f*g)_{1}&=&f_{0}g_{1}+f_{1}g_{0}+f_{2}g_{3}-f_{3}g_{2}\\\
(f*g)_{2}&=&f_{0}g_{2}-f_{1}g_{3}+f_{2}g_{0}+f_{3}g_{1}\\\
(f*g)_{3}&=&f_{0}g_{3}+f_{1}g_{2}-f_{2}g_{1}+f_{3}g_{0}F\end{array}$

We now set

$f^{s}:=f_{0}^{2}+f_{1}^{2}+f_{2}^{2}+f_{3}^{2}=f*f^{c}=f^{c}*f$

and call $f^{s}$ the _symmetrization_ of $f$. Let us mention that the equality
(2.1) immediately implies $|f_{0}(z)|\leq|f(z)|$ and replacing $f$ by
$f*\iota$ for $\iota=i,j,k$ gives also the remaining inequalities
$|f_{l}(z)\leq|f(z)|,\,l=1,2,3.$

It is known that Runge theorem holds for slice–preserving functions and for
slice–regular functions. In order to prove the Runge theorem also for
$\mathcal{SR}_{\omega}(\Omega)$ (Theorem 2.12), we will not only use Theorem
4.10, but also the vanishing of $H^{1}(\Omega,\mathcal{SR}_{\mathbb{R}}^{*})$
(Theorem 4.2 (1)) which follows from the vanishing of
$H^{1}(\Omega,\mathcal{SR}_{\mathbb{R}})$ (Theorem 4.1 (1)).

######  Theorem 2.12 (Runge Theorem).

Let $U\subset\Omega$ be axially symmetric domains, the set $U_{I}$ Runge in
$\Omega_{I}$ for some $I\in\mathbb{S}$ and $\omega$ a vectorial class on
$\Omega$. Then Runge theorem holds in classes (1)
$\mathcal{SR}_{\mathbb{R}}(\Omega),$ (2) $\mathcal{SR}_{\omega}(\Omega)$ and
(3) for general slice–regular functions.

The proofs of (1) and (3) follow from classical complex analysis results.
Runge theorem holds also for nonvanishing slice–regular functions.

######  Theorem 2.13 (Runge theorem for $\mathcal{SR}^{*}$, Theorem 3.8
[PV2]).

Let $K$ be a union of finitely many disjoint closed basic sets in $\mathbb{H}$
and let $f\in\mathcal{SR}^{*}(\Omega)$ with $\Omega$ an open axially symmetric
neighborhood of $K$. Then, for any $\varepsilon>0$, there exists
$g\in{\mathcal{SR}^{*}}(\mathbb{H})$ such that

$|f-g|_{K}<\varepsilon.$

######  Remark 2.14.

Any finite union $K$ of disjoint closed basic sets is Runge in any axially
symmetric open set $\Omega\supset K$.

######  Remark 2.15.

Runge Theorem 2.13 holds also if the group $\mathcal{SR}^{*}(\Omega)$ is
replaced by some other closed and connected subgroup, where Runge theorem is
available, for example the subgroup $\mathcal{SR}^{*}_{\omega}(\Omega).$

###  2.1. Matrix representations

As complex numbers can be represented by matrices, so can be quaternions.
There are at least two ways of representing quaternions as matrices in such a
way that quaternion addition and multiplication correspond to matrix addition
and matrix multiplication. One is to use $2\times 2$ complex matrices, and the
other is to use $4\times 4$ real matrices.

In the terminology of abstract algebra, these are injective homomorphisms from
$\mathbb{H}$ to the matrix rings $M(2,\mathbb{C})$ and $M(4,\mathbb{R})$,
respectively. Furthermore, there is a strong relation between quaternion units
and Pauli matrices $\sigma_{1},\sigma_{2},\sigma_{3}$ (which are $2\times 2$
complex Hermitian unitary matrices and appear in quantum mechanics) since the
real linear span of $\\{\operatorname{Id}_{2},\ i\sigma_{1},\ i\sigma_{2},\
i\sigma_{3}\\}$ is isomorphic to the real algebra of quaternions.

Our aim is to provide any slice function with a matrix representation whose
entries are slice-preserving functions in such a way that the $*$–product of
slice functions will correspond to the matrix product of corresponding
functions in the same spirit as [AdF3].

To this end we identify

$\mathbf{1}=\operatorname{Id}_{4}=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{bmatrix}\leftrightarrow
1\quad\mathbf{I}=\begin{bmatrix}0&-1&0&0\\\ 1&0&0&0\\\ 0&0&0&-1\\\
0&0&1&0\end{bmatrix}\leftrightarrow i$

$\mathbf{J}=\begin{bmatrix}0&0&-1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\
0&-1&0&0\end{bmatrix}\leftrightarrow
j\quad\mathbf{K}=\begin{bmatrix}0&0&0&-1\\\ 0&0&-1&0\\\ 0&1&0&0\\\
1&0&0&0\end{bmatrix}\leftrightarrow k$

since $\mathbf{I}^{2}=\mathbf{J}^{2}=\mathbf{K}^{2}=-\mathbf{1}$ and
$\mathbf{I}\cdot\mathbf{J}=\mathbf{K}$ and so

$f\leftrightarrow
f_{0}\mathbf{1}+f_{1}\mathbf{I}+f_{2}\mathbf{J}+f_{3}\mathbf{K}\ \mathrm{or}\
f\leftrightarrow\begin{bmatrix}f_{0}&-f_{1}&-f_{2}&-f_{3}\\\
f_{1}&f_{0}&-f_{3}&f_{2}\\\ f_{2}&f_{3}&f_{0}&-f_{1}\\\
f_{3}&-f_{2}&f_{1}&f_{0}\end{bmatrix}=:M_{f}.$

Thus, if analogously $g\leftrightarrow
M_{g}=\begin{bmatrix}g_{0}&-g_{1}&-g_{2}&-g_{3}\\\ g_{1}&g_{0}&-g_{3}&g_{2}\\\
g_{2}&g_{3}&g_{0}&-g_{1}\\\ g_{3}&-g_{2}&g_{1}&g_{0}\end{bmatrix}$, then

$f*g\leftrightarrow M_{f*g}=M_{f}\cdot M_{g}$

as desired (see also (2.3)). Notice furthermore that $f^{c}\leftrightarrow
M_{f}^{T}$, where $M_{f}^{T}$ is the transpose of $M_{f}$, and so

$M_{f^{s}}=M_{f*f^{c}}=M_{f}\cdot M_{f}^{T}=\begin{bmatrix}f^{s}&0&0&0\\\
0&f^{s}&0&0\\\ 0&0&f^{s}&0\\\ 0&0&0&f^{s}\end{bmatrix}=f^{s}\mathbf{1}.$

Moreover, if $f^{s}\not\equiv 0$, then $f^{-*}:=f^{c}/f^{s}$ is such that
$f^{-*}*f=f*f^{-*}=\operatorname{Id}$, that is

$M_{f^{-*}}\cdot M_{f}=M_{f}\cdot M_{f^{-*}}=\mathbf{1}.$

Because $\det M_{f}=(f^{s})^{2}$ we observe, as expected, that $f$ has a
$*$-inverse if and only if $f^{s}$ is not vanishing, i.e. $M_{f}$ is an
invertible (or nonsingular) matrix.

Let $\iota\in\mathbb{S}$ and $\mathbb{C}_{\iota}$ the induced complex plane.
Define $\mathcal{M}_{\mathbb{C}_{\iota}}$ to be the set of matrices

$\mathcal{M}_{\mathbb{C}_{\iota}}=\\{M=q_{0}\mathbf{1}+q_{1}\mathbf{I}+q_{2}\mathbf{J}+q_{3}\mathbf{K},q_{0},\ldots,q_{3}\in\mathbb{C}_{\iota}\\}$

and equip it with the operator norm
$|A|_{\infty}:=\max_{i=0,..,3}\sum_{j=0}^{3}|a_{ij}|.$

Because for our class of matrices the sums of absolute values of rows are the
same, the induced norm satisfies

$|M|=\sum_{j=0}^{3}|q_{j}|=|M^{T}|=|M|_{\infty}=|M|_{1}.$

Notice that
$\mathcal{M}_{\mathbb{R}}=\\{r_{0}\mathbf{1}+r_{1}\mathbf{I}+r_{2}\mathbf{J}+r_{3}\mathbf{K},r_{0},\ldots,r_{3}\in\mathbb{R}\\}\simeq\mathbb{H}.$

######  Definition 2.16.

The set of all matrices $M_{f}$ with entries slice–preserving slice functions
defined in $\Omega\subseteq\mathbb{H}$ is denoted by
$\mathcal{M}_{\mathcal{S}_{\mathbb{R}}}(\Omega)$ and the subgroup of non
singular ones by $\mathcal{GL}_{\mathcal{S}_{\mathbb{R}}}(\Omega)$. The
subgroup of matrices with slice–regular slice–preserving functions is denoted
by $\mathcal{M}_{\mathcal{SR}_{\mathbb{R}}}(\Omega)$ and the subgroup of
invertible ones by $\mathcal{GL}_{\mathcal{SR}_{\mathbb{R}}}(\Omega).$

Notice that if $z\in\Omega_{I}\setminus\mathbb{R}$, then
$M_{f}(z)\in\mathcal{M}_{\mathbb{C}_{I}}$ and for $z\in\Omega\cap\mathbb{R}$
we have $M_{f}(z)\in\mathcal{M}_{\mathbb{R}.}$ The inequalities
$|f(z)|\leq|M_{f}(z)|\leq 4|f(z)|$ hold and in addition
$|M_{f}(z)|=|M_{f^{c}}(z)|.$ We equip the set
$\mathcal{M}_{\mathcal{S}_{\mathbb{R}}}(\Omega)$ with the sup norm

$|M_{f}|_{\Omega}:=\sup_{z\in\Omega}|M_{f}(z)|.$

The subaditivity, submultiplicativity and homogeneity of this norm are
evident.

As customary, one puts, $M_{f}^{n}:=M_{f}\cdot M_{f}^{n-1}$ with
$M_{f}^{0}:=\mathbf{1}$; therefore, one can define

$\exp(M_{f}):=\sum_{n=0}^{+\infty}\dfrac{M_{f}^{n}}{n!}\,\mbox{ and
}\,\log({\mathbf{1}}-M_{f}):=-\sum_{n=1}^{+\infty}\dfrac{M_{f}^{n}}{n}$

and observe that

$\exp{M_{f}}\leftrightarrow\exp_{*}(f)\,\mbox{ and
}\,\log({\mathbf{1}}-M_{f})\leftrightarrow\log_{*}(1-f)$

as introduced in [AdF1, AdF2, GPV]. Notice that the series for exponential
function converges for any matrix $M_{f}$ whereas the series for logarithm
converges in $\Omega$ for all matrices $M_{f}$ satisfying
$|M_{f}|_{\Omega}<1$. Since $\mathbf{1}$ commutes with $M_{f}^{n}$ (for any
$n$), one easily deduces that

$\exp(\log(\mathbf{1}-M_{f}))=\mathbf{1}-M_{f},\quad\exp(M_{f})\cdot\exp(-M_{f})=\mathbf{1}.$

The second equality proves that $\exp(M_{f})$ is nonsingular for any matrix
$M_{f}$. From the first equality, if $|\mathbf{1}-M_{f}|<1$, then
$M_{f}=\log(M_{g})$ with $M_{g}=\exp(M_{f})$.

Assume that $f=f_{0}+f_{v}=f_{0}+f_{1}v\in\mathcal{SR}_{[v]}(U)$ with $U$ an
axially symmetric open set and $v^{2}=-v^{s}=-1$,
$f_{0},f_{1}\in\mathcal{SR}_{\mathbb{R}}(U)$. Then $v$ is a minimal
representative of $[v]$ and we can represent $f$ by a matrix

$M_{f}:=\begin{bmatrix}f_{0}&-f_{1}\\\ f_{1}&f_{0}\end{bmatrix}.$

Moreover, $\det M_{f}=f^{s},$ so the matrix is invertible if and only if $f$
is nonzero. If also $g\in\mathcal{SR}_{[v]}(U),$ then
$M_{f*g}=M_{f}M_{g}=M_{g}M_{f}$, which implies that $\exp(M_{f}+M_{g})=(\exp
M_{f})(\exp M_{g})$.

##  3\. Sheaf theory for slice–regular functions

The notion of a sheaf of slice–regular functions was already studied in the
paper [CSS]. Here we give a slightly different definition of sheaves.

Following Propositions $4.2$ and $4.3$ in [CSS] we have

######  Proposition 3.1.

The assignment
$\\{U,\mathcal{S}\mathcal{R}(U)\\}_{U=\widetilde{U}\subset\mathbb{H}}$ of
rings of slice–regular functions over all axially symmetric open sets $U$ in
$\mathbb{H}$ together with the families of morphisms
$\rho_{U,V}:\mathcal{SR}(U)\rightarrow\mathcal{SR}(V)$ for every $V\subset U,$
$U,V$ axially symmetric, induced by restrictions, is a presheaf. The presheaf
$\\{U,\mathcal{S}\mathcal{R}(U)\\}_{U=\widetilde{U}\subset\mathbb{H}}$ is a
sheaf denoted by $\mathcal{SR}$.

The same holds true for the assignments
$\\{U,\mathcal{SMR}(U)\\}_{U=\widetilde{U}\subset\mathbb{H}}$ and
$\\{U,\mathcal{S}(U)\\}_{U=\widetilde{U}\subset\mathbb{H}}$ over all $U$
axially symmetric open sets. The corresponding sheaves are denoted by
$\mathcal{SMR}$ and $\mathcal{S}$ respectively.

In this manner, the stalks of the sheaf are germs of functions defined on the
elements of the set $\mathcal{H}:=\\{\widetilde{z},\,z\in\mathbb{H}\\}$ and
denoted by $f_{*,\widetilde{z}}$. Notice that $\mathcal{H}=\mathbb{H}/\sim,$
where the relation is $z\sim w\Leftrightarrow\,\exists\,a,r\in\mathbb{R}\mbox{
so that }z,w\in S(a,r)$. Recall that the symmetrization of any real point $x$
is the set $\widetilde{x}=\\{x\\}$. By a slight abuse of notation, in such a
case, we simply write $x$.

######  Remark 3.2.

Let us explain what is the main difference between the definition of sheaves
in [CSS] and the one in Proposition 3.1. In [CSS] the authors have supposed
that the open sets are arbitrary, but the functions considered were assumed to
be extendable to the symmetrization of the open set where they are defined.
The problem that we encountered with this approach was, that when we have
three non axially symmetric sets $U_{1},U_{2},U_{3},$ such that $U_{1}\cap
U_{2}\cap U_{3}:=U_{123}=\varnothing$ and such that their symmetrizations
$\tilde{U}_{1},\tilde{U}_{2},\tilde{U}_{3},$ are the same set $U,$ then any
choice of functions $f_{lm}$ in $U_{lm}$ which extend to the symmetrization of
$U_{lm}$, satisfying only the condition $f_{lm}=-f_{ml},$ $1\leq l<m\leq 3$,
forms a cocycle. While the cocycle condition $f_{123}=f_{12}+f_{23}+f_{31}$ is
trivial on the sets $U_{123}=\varnothing$, this is no longer true so for the
corresponding symmetrized sets. Therefore the cocycles do not ‘symmetrize’ to
cocycles.

Let us make an explicit example. Assume
$U_{0}:=\\{q\in\mathbb{H},d(q,\mathbb{S})<1/2\\}.$ Let $C$ be the circle
$i\cos t+j\sin t,t\in[0,2\pi].$ Divide the circle into arcs
$C_{l}:=[(l-1)2\pi/2,l2\pi/3],l=1,2,3$ and let $S_{l}$ be tubular
neighbourhoods of $C_{l}$ in $\mathbb{S}$ such that $S_{1}\cap S_{2}\cap
S_{3}=\varnothing.$ Since the set $U_{0}$ does not intersect the real axis,
the imaginary unit function $\mathcal{I}$ is well–defined on $U_{0}$ and hence
the sets $U_{l}=\\{q\in U_{0},{\rm Im}{q}\in S_{l}\\},l=1,2,3$ form an open
covering of $U:=U_{1}\cup U_{2}\cup U_{3}$. By construction,
$U_{123}=\varnothing$. Moreover, the symmetrization of each of the sets
$U_{1},U_{1},U_{3}$ is $U_{0}.$ Set $f_{lm}=1,$ $f_{mm}=0$ and
$f_{lm}=-f_{ml},1\leq l<m\leq 3$ and denote the symmetrizations by
$\tilde{f}_{lm}.$ Then $\tilde{f}_{12}+\tilde{f}_{23}+\tilde{f}_{31}=1\neq 0.$
We claim that $H^{1}(U,\mathcal{SR})\neq 0$ in the sense of [CSS]. If it were
zero, then we would have functions $f_{l}$ on $U_{l}$ with
$f_{lm}=f_{m}-f_{l}=1,l<m$ and hence, by the identity principle, the same
would hold for the symmetrized functions $\tilde{f}_{l},$
$\tilde{f}_{lm}=\tilde{f}_{m}-\tilde{f}_{l}=1,l<m.$ But then

$1=\tilde{f}_{12}+\tilde{f}_{23}+\tilde{f}_{31}=\tilde{f}_{2}-\tilde{f}_{1}+\tilde{f}_{3}-\tilde{f}_{2}+\tilde{f}_{1}-\tilde{f}_{3}=0,$

which is a contradiction. To put it shortly, the symmetrization process and
cocycle condition are not commutative operations.

With the definition we have adopted this problem does not occur. At first
glance the requirement for symmetric coverings seems to be a restriction since
the sets we are dealing with are not small in diameter. On the other hand, the
set of functions considered in [CSS] is also restricted by the requirement
that they are extendable to the symmetrized set. On the level of presheaves
both definitions are about the same set of functions. The difference occurs on
the level of cocycles.

Recall that $\mathcal{SR}_{\mathbb{R}}=\mathcal{SR}_{[0]}$ and
$\mathcal{SR}_{\omega}$ is an $\mathcal{SR}_{\mathbb{R}}$-module. The sheaves
$\mathcal{SR}_{\mathbb{R}}$ and $\mathcal{SR}_{\omega}$ are subsheaves of
$\mathcal{SR}$. All these three sheaves are also sheaves of abelian groups for
addition. Notice that
$\mathcal{SR}_{\mathbb{R}}\subset\mathcal{SR}_{\omega}\subset\mathcal{SR}$ for
any $\omega.$

In addition, we define
$\mathcal{SR}_{\mathbb{R}}^{+}\subset\mathcal{SR}_{\mathbb{R}}$ to be the
subset of germs $f_{\widetilde{z}}\in\mathcal{SR}_{\mathbb{R}}$ which at the
point $z_{0}\in\mathbb{R}$ satisfy the condition

(3.4) $f_{*,z_{0}}(z_{0})\in(0,\infty).$

Also $\mathcal{SR}_{\mathbb{R}}^{+}$ is a subsheaf of
$\mathcal{SR}_{\mathbb{R}}$.

As customary, by $\mathcal{SR}^{*}$ we denote the set of all nonvanishing
germs $f_{*}\in\mathcal{SR}$ and define
$\mathcal{SR}_{\mathbb{R}}^{*,+}:=\mathcal{SR}^{+}_{\mathbb{R}}\cap\mathcal{SR}^{*}$,
$\mathcal{SR}_{\mathbb{R}}^{*}:=\mathcal{SR}_{\mathbb{R}}\cap\mathcal{SR}^{*}$,
$\mathcal{SR}_{\omega}^{*}:=\mathcal{SR}_{\omega}\cap\mathcal{SR}^{*}.$ The
sheaves $\mathcal{SR}_{\mathbb{R}}^{*}$ and $\mathcal{SR}_{\omega}^{*}$ are
sheaves of abelian groups for $*$-multiplication. We also set
$\mathcal{SMR}^{*}:=\mathcal{SMR}\setminus\\{0\\}.$

######  Definition 3.3.

In $\mathcal{SMR}$ two germs $f_{*},g_{*}\in\mathcal{SMR}$ are equivalent,
$f_{*}\sim g_{*}$, if there exists $h_{*}\in\mathcal{SR}^{*}$ so that
$f_{*}=g_{*}*h_{*}.$

A divisor on an axially symmetric domain $\Omega$ is an element in
$\Gamma(\Omega,\mathcal{SMR}^{*}/\sim).$

Given a divisor on an axially symmetric domain $\Omega$, there exists a basic
covering $\\{U_{l}\\}_{l\in\mathbb{N}}$ of $\Omega$ so that the divisor is
represented by the pairs $\\{(f_{l},U_{l}),\
f_{l}\in\mathcal{SMR}(U_{l})\\}_{l\in\mathbb{N}}$ satisfying
$f_{kl}=f_{k}^{-*}*f_{l}\in\mathcal{SR}^{*}(U_{kl}),k,l\in\mathbb{N}.$

######  Remark 3.4.

The support of a divisor on an axially symmetric domain $\Omega$ is a
(discrete) set of points and spheres in $\Omega$.

We recall the following (see [GPV]).

######  Proposition 3.5.

Let $\omega=[v]$ be a vectorial class on an axially symmetric domain $\Omega$
such that $v^{s}\neq 0.$ The exponential maps between sheaves induced by
$\exp_{*},$

(3.5) $\displaystyle\mathcal{E}_{*}:\mathcal{SR}\rightarrow\mathcal{SR}^{*}$
$\displaystyle\mathcal{E}_{*}:\mathcal{SR}_{\omega}\rightarrow\mathcal{SR}_{\omega}^{*}$
$\displaystyle\mathcal{E}_{*}:\mathcal{SR}_{\mathbb{R}}\rightarrow\mathcal{SR}_{\mathbb{R}}^{*,+}$

are surjective and the latter two are also sheaf homomorphisms.

If $v$ is a minimal representative of $\omega$ on a basic neighbourhood $U$ of
$z$ in $\Omega$, then the kernels ${\mathcal{K}}_{\omega}$ and
${\mathcal{K}}_{\mathbb{R}}$ have the following local description:

  * i)

if $z\not\in\mathbb{R}$ and $v^{s}(z)\neq
0\Rightarrow{\mathcal{K}}_{\omega,z}=\\{(2k+\delta)\pi\mathcal{I}+(2n+\delta)\pi\frac{v}{\sqrt{v^{s}}},\,k,n\in\mathbb{Z},\delta=0,1\\},$

  * ii)

if $z\in\mathbb{R}$ and $v^{s}(z)\neq
0\Rightarrow{\mathcal{K}}_{\omega,z}=\\{2n\pi\frac{v}{\sqrt{v^{s}}},\,n\in\mathbb{Z}\\},$

  * iii)

if $z\not\in\mathbb{R}$ and
$v^{s}(z)=0\Rightarrow{\mathcal{K}}_{\omega,z}=\\{2k\pi\mathcal{I},\,k\in\mathbb{Z}\\},$

  * iv)

if $z\not\in\mathbb{R}$,
${\mathcal{K}}_{\mathbb{R},z}=\\{2k\pi\mathcal{I},\,k\in\mathbb{Z}\\}$ and if
$z\in\mathbb{R}$, ${\mathcal{K}}_{\mathbb{R},z}=\\{0\\}$.

Since $\mathcal{SR}=\cup_{\omega}\mathcal{SR}_{\omega}$ we have for each
$z\in\Omega$

${\mathcal{K}}_{z}:=\mathcal{E}_{*}^{-1}(1)=\cup_{{\omega}}{\mathcal{K}}_{\omega,z},$

with $\mathcal{E}_{*}^{-1}(1)$ denoting the inverse image of $1$ in (3.5).

######  Theorem 3.6.

The sequences

$0\rightarrow{\mathcal{K}}_{\omega}\rightarrow\mathcal{SR}_{\omega}\xrightarrow{\mathcal{E}_{*}}\mathcal{SR}_{\omega}^{*}\rightarrow
0$
$0\rightarrow{\mathcal{K}}_{\mathbb{R}}\rightarrow\mathcal{SR}_{\mathbb{R}}\xrightarrow{\mathcal{E}_{*}}\mathcal{SR}_{\mathbb{R}}^{*,+}\rightarrow
0$

are exact.

######  Remark 3.7.

The sequence

$0\rightarrow{\mathcal{K}}\rightarrow\mathcal{SR}\xrightarrow{\mathcal{E}_{*}}\mathcal{SR}^{*}\rightarrow
0$

is not exact. If $f\in\mathcal{SR}_{\omega}(\Omega)$ and
$g\in\mathcal{SR}_{\omega_{1}}(\Omega)$, $\omega\neq\omega_{1},$ then
$\exp_{*}(f+g)\neq\exp_{*}f*\exp_{*}g$ in general. In particular, if $f=2\pi
i,$ $g=2\pi j$, then $\exp_{*}f=\exp_{*}g=1$ but

$\exp_{*}(f+g)=\cos(2\sqrt{2}\pi)+\frac{i+j}{\sqrt{2}}\sin(2\sqrt{2}\pi)\neq
1.$

We begin by studying the Čech cohomology of $\Omega$ with values in the
abelian group $\mathcal{I}\mathbb{Z}$ with respect to axially symmetric
coverings.

######  Definition 3.8.

Let $\Omega\subset\mathbb{H}$ be an axially symmetric domain in $\mathbb{H}$
and $\mathcal{U}=\\{U_{\lambda}\\}_{\lambda\in\Lambda}$ an axially symmetric
open covering of $\Omega$. An $m$-cochain of $\mathcal{U}$ with coefficients
in $\mathcal{I}\mathbb{Z}$ is any collection
$\\{(f_{L},U_{L}),f_{L}=\mathcal{I}n_{L},\
n_{L}\in\mathbb{Z}\\}_{L\in\Lambda^{m}}$ with $f_{L}$ continuous.

Notice that on a slice domain the function $\mathcal{I}n_{L}$ is continuous if
and only if $n_{L}=0$. Cochain complexes and coboundary operators are defined
as usual and corresponding cohomology groups denoted by
$H^{n}(\Omega,\mathcal{I}\mathbb{Z})$.

######  Theorem 3.9.

Let $\Omega$ be an axially symmetric domain. Then

$H^{n}(\Omega,\mathcal{I}\mathbb{Z})=0\mbox{ for }n\geq 2.$

We prove the theorem by restricting the data on a slice. This induces
antisymmetric complexes introduced in [PV2].

######  Definition 3.10 (Antisymmetric complexes and cohomology groups).

Let $D\subset\mathbb{C}$ be an open symmetric set and
$\mathcal{U}=\\{U_{\lambda}\\}_{\lambda\in\Lambda}$ an open symmetric covering
of $D$. An antisymmetric $m$-cochain of $\mathcal{U}$ is any collection
$\\{(f_{L},U_{L}),f_{L}:U_{L}\rightarrow\mathbb{Z}\\}_{L\in\Lambda^{m}}$ with
$f_{L}$ continuous, constant in $U_{L}^{+}$ and satisfying the antisymmetric
property:

$f_{L}(z)=-f_{L}(\bar{z}).$

Antisymmetric cochain complexes and coboundary operators are defined as usual
and corresponding cohomology groups denoted by
$H^{n}_{a}(\mathcal{U},\mathbb{Z})$. The open symmetric coverings of $D$ form
a directed set under refinement, therefore we define $H^{n}_{a}(D,\mathbb{Z})$
to be a direct limit over symmetric open coverings of $D$.

In [PV2], Theorem 5.2 we have proved that for any axially symmetric domain
$\Omega\subset\mathbb{H}$ we have $H^{n}_{a}(\Omega_{I},\mathbb{Z})=0$ for all
$n\geq 2$.

######  Proposition 3.11.

Let $\Omega$ be an axially symmetric domain in $\mathbb{H}$. By restricting to
$\Omega_{I},$ any cochain on $\Omega$ with values in $\mathcal{I}\mathbb{Z}$
induces an antisymmetric cochain on $\Omega_{I}$. Vice versa, by
symmetrization of domains, every antisymmetric cochain on $\Omega_{I}$ induces
a cochain on $\Omega$ with values in $\mathcal{I}\mathbb{Z}$.

###### Proof.

Consider an $m$-cochain defined on an axially symmetric open covering
$\mathcal{U}=\\{U_{n}\\}_{n\in\mathbb{N}_{0}}$ of $\Omega$ by
$C=\\{({\mathcal{I}}n_{L},U_{L})\\}_{L\in\mathbb{N}_{0}^{m}}$, where $n_{L}=0$
if $U_{L}\cap\mathbb{R}=\varnothing$. Choose $I\in\mathbb{S}$ and consider the
covering $\mathcal{U}_{I}=\\{U_{n,I}\\}_{n\in\mathbb{N}_{0}}$ of the symmetric
set $\Omega_{I}\subset\mathbb{C}_{I}$. Notice that if the open set $U_{k,I}$
intersects the real axis, it is a topological disc with $R(U_{k,I})=U_{k,I}$
and hence $n_{L}=0$ for all $L$ that contain $k$. If it does not intersect the
real axis then $U_{k,I}$ consists of two symmetric topological discs
$U_{k,I}^{+}$ and $U_{k,I}^{-}.$ Hence $C$ equals $In_{L}$ on $U_{L}^{+}$ and
$-In_{L}$ on $U_{L}^{-}$ and so assigns the value $n_{L}$ on $U_{L}^{+}$ and
the value $-n_{L}$ on $U_{L}^{-}.$ The reverse and the last statement are
obvious.

∎

###### Proof of Theorem 3.9.

The proof is a direct consequence of Proposition 3.11 and the fact that
$H^{n}_{a}(\Omega_{I},\mathbb{Z})=0$ for all $n\geq 2$. ∎

##  4\. Cousin problems and Cartan’s Splitting Lemmas for Classes of Slice
Regular Functions

This section is devoted to the proofs of the following two theorems and their
applications.

######  Theorem 4.1.

Let $\Omega$ be an axially symmetric domain in $\mathbb{H}$ and $\omega$ a
global vectorial class on $\Omega$. Then

  * 1)

$H^{1}(\Omega,\mathcal{SR}_{\mathbb{R}})=0$;

  * 2)

$H^{1}(\Omega,\mathcal{SR}_{\omega})=0$;

  * 3)

$H^{1}(\Omega,\mathcal{SR})=0.$

This implies that the first (or additive) Cousin problem is solvable in
$\mathcal{SR}_{\mathbb{R}}$, in $\mathcal{SR}_{\omega}$ and in $\mathcal{SR}$.

######  Theorem 4.2.

Let $\Omega$ be an axially symmetric domain in $\mathbb{H}$ and $\omega$ a
global vectorial class on $\Omega$. Then

  * 1)

$H^{1}(\Omega,\mathcal{SR}^{*,+}_{\mathbb{R}})=H^{1}(\Omega,\mathcal{SR}_{\mathbb{R}}^{*})=0$;

  * 2)

$H^{1}(\Omega,\mathcal{SR}^{*}_{\omega})=0$;

  * 3)

$H^{1}(\Omega,\mathcal{SR}^{*})=0.$

This implies that the second (or multiplicative) Cousin problem is solvable in
$\mathcal{SR}^{*}_{\mathbb{R}}$, in $\mathcal{SR}^{*}_{\omega}$ and in
$\mathcal{SR}^{*}$.

Theorem 1.1 is a special case of Theorem 4.2. The claim in Theorem 4.1 (3) is
analogous to the one of Proposition $4.8$ in [CSS].

To have a concise presentation, the analogous results for all the three
classes of functions are stated in one theorem. However, the proofs of all the
three claims are based on the proofs for the class of slice–preserving
functions. In particular, Theorem 4.1 (1) implies Theorem 4.1 (3) and Theorem
4.2 (1). Theorem 4.2 (1) implies Theorem 4.10 on existence of global minimal
representatives of the vectorial class $\omega$ on an axially symmetric open
set $\Omega$ and consequently the Runge theorem 2.12 holds in
$\mathcal{SR}_{\omega}$. Theorem 4.2 (1) and Theorem 4.10 then imply Theorem
4.1 (2).

###  4.1. Additive and multiplicative Cartan’s Splitting lemmas

We begin this subsection with the proofs of additive and multiplicative
Cartan’s Splitting Lemmas for slice–preserving functions, functions with
prescribed vectorial classes and for general slice–regular functions. The
reason why these classes are treated separately is that the $*$-product is
commutative in the first two classes.

We recall the notion of Cartan pairs and strings from Section 4, [PV2]. If
$A,B\subset\mathbb{H}$ are axially symmetric compact sets with piecewise
smooth boundaries satisfying $\overline{(A\setminus
B)}\cap\overline{(B\setminus A)}=\varnothing$, $B$ a closed basic set and
$A\cap B$ a finite union of disjoint closed basic sets, then we say that
$(A,B)$ form a Cartan pair or a Cartan $2$-string. A sequence
$A_{1},\ldots,A_{n}\subset\mathbb{H}$ of axially symmetric compact sets with
piecewise smooth boundary is a Cartan $n$-string if $(A_{1},\ldots,A_{n-1})$
and $(A_{1}\cap A_{n},\ldots A_{n-1}\cap A_{n})$ are Cartan $(n-1)$-strings
and $(A_{1}\cup\ldots\cup A_{n-1},A_{n})$ is a Cartan pair.

A sequence $\mathcal{A}=\\{A_{n}\\}_{n\in\mathbb{N}_{0}}$ of closed basic sets
is a Cartan sequence if for each $n\in\mathbb{N}$, then $(A_{0},\ldots,A_{n})$
is a Cartan $n+1$-string and for distinct $n_{1},n_{2},n_{3},n_{4}$, we have
that $A_{n_{1}n_{2}n_{3}n_{4}}=\varnothing$ and $A_{n_{1}n_{2}n_{3}}$,
$A_{n_{1}n_{2}}$ are closed basic sets if nonempty; if
$A_{n_{i}}\cap\mathbb{R}\neq\varnothing$ for distinct $n_{1},n_{2},n_{3}$,
then $A_{n_{1}n_{2}n_{3}}=\varnothing$ and the set $A_{n_{1}n_{2}}$ is a slice
basic domain if nonempty.

A covering $\mathcal{B}=\\{U_{k}\\}_{k\in\mathbb{N}_{0}}$ is a Cartan covering
of $\Omega$ if it is a covering of $\Omega$ and the sequence
$\overline{U}_{0},\overline{U}_{1},\ldots$ is a Cartan sequence.

Theorem 1.1 in [PV2], states that given ${\mathcal{U}}$ a locally finite
axially symmetric open covering of an axially symmetric domain
$\Omega\subset\mathbb{H}$ and $Z\subset\Omega$, a discrete set of points or
spheres, there exist a Cartan covering
$\mathcal{B}=\\{U_{n}\\}_{n\in\mathbb{N}_{0}}$ subordinated to
$({\mathcal{U}},Z).$ Even more, there exist a sequence
$\\{\varepsilon_{n}\\}_{n\in\mathbb{N}_{0}}$ such that also the coverings

${\mathcal{B}}^{t}:=\\{U_{n}^{t}=U_{n}+B(0,t\varepsilon_{n})\\}_{n\in\mathbb{N}_{0}}$

are Cartan and subordinated to $({\mathcal{U}},Z)$ for all $t\in[0,1].$ In
particular, $\overline{U}_{n}=\cap_{t>0}U^{t}_{n}.$

######  Lemma 4.3 (Additive Cartan’s splitting lemma with estimates).

Let $(A,B)$ be a Cartan pair, $C:=A\cap B$ and $U$ its open neighbourhood. Let
$U_{A}\Subset\mathbb{H}$ be a (topologically) regular axially symmetric open
neighbourhood of $A$, $U_{B}\Subset\mathbb{H}$ a (topologically) regular
axially symmetric open neighbourhood of $B$ so that
$(\overline{U_{A}},\overline{U_{B}})$ is a Cartan pair with $U_{C}:=U_{A}\cap
U_{B}\Subset U.$

There exists a constant $D$, so that any $\gamma\in\mathcal{SR}(U)$ can be
decomposed as $\gamma=\alpha+\beta$ on $U_{C}$ with
$\alpha\in\mathcal{SR}(U_{A}),$ $\beta\in\mathcal{SR}(U_{B}),$ satisfying the
estimate

(4.6) $|\alpha|_{A},|\beta|_{B}\leq D|\gamma|_{\overline{U_{C}}}.$

In addition,

(a) if $\gamma$ is slice–preserving, then we can choose $\alpha$ and $\beta$
to be slice–preserving;

(b) if $\omega$ is a vectorial class on $\Omega\Supset(U\cup U_{A}\cup
U_{B})$, $g_{v}$ its global minimal representative on $\Omega$, $g_{v}\neq 0$
on $U$, and if $\gamma\in\mathcal{SR}_{\omega}(U),$ then we can choose
$\alpha\in\mathcal{SR}_{\omega}(U_{A})$ and
$\beta\in\mathcal{SR}_{\omega}(U_{B}).$

###### Proof.

The proof is divided in three parts.

Proof for $\mathcal{SR}_{\mathbb{R}}$. We claim that there exist functions
$\alpha\in\mathcal{SR}_{\mathbb{R}}(U_{A})$,
$\beta\in\mathcal{SR}_{\mathbb{R}}(U_{B}),$ $\gamma=\alpha+\beta$ and such
that the estimate (4.6) holds. Fix $I\in\mathbb{S}$. By classical results
(compare [GR], VI.E, Lemma 6), there exist holomorphic functions
$\alpha_{1}\in\mathcal{O}(U_{A,I})$ and $\beta_{1}\in\mathcal{O}(U_{B,I})$ so
that $\gamma_{I}=\alpha_{1}+\beta_{1}$ satisfying the estimate
$|\alpha_{1}|_{A},|\beta_{1}|_{B}\leq D|\gamma_{I}|_{\overline{U_{C,I}}}$.
Define $\alpha_{I}$ by
$\alpha_{I}(z)=(\alpha_{1}(z)+\overline{\alpha_{1}(\bar{z})})/2$ and
$\beta_{I}$ by $\beta_{I}(z)=(\beta_{1}(z)+\overline{\beta_{1}(\bar{z})})/2$.
Then $|\alpha_{I}|_{A},|\beta_{I}|_{B}\leq
D|\gamma_{I}|_{\overline{U_{C,I}}}$.

By extension formula, the functions $\alpha_{I}$ and $\beta_{I}$ extend to
slice–preserving functions $\alpha,\beta$. Then, by construction, the equality
$\gamma=\alpha+\beta$ and the estimate (4.6) hold.

Proof for $\mathcal{SR}_{\omega}$. Set $M:=|g_{v}|_{\overline{U_{A}\cup
U_{B}}}$ and $m:={\rm min}\,\\{|g_{v}(q)|,q\in\overline{U_{C}}\\}.$ Decompose
the function $\gamma=\gamma_{0}+\gamma_{v}.$ Since $g_{v}$ is a minimal
representative of the vectorial class $\omega$, we have
$\gamma_{v}=\gamma_{1}g_{v}\mbox{ with
}\gamma_{1}\in\mathcal{SR}_{\mathbb{R}}$. A trivial estimate
$|\gamma_{1}|_{\overline{U_{C}}}\leq|\gamma_{v}|_{\overline{U_{C}}}/m$ holds.
For $l=0,1,$ the functions $\gamma_{l}$ can be decomposed to
$\gamma_{l}=\alpha_{l}+\beta_{l},$ with
$\alpha_{l}\in\mathcal{SR}_{\mathbb{R}}({U}_{A}),$
$\beta_{l}\in\mathcal{SR}_{\mathbb{R}}({U}_{B})$ with the estimate

(4.7) $|\alpha_{l}|_{A},|\beta_{l}|_{B}\leq
D_{1}|\gamma_{l}|_{\overline{U_{C}}}.$

Define
$\alpha:=\alpha_{0}+\alpha_{1}g_{v},\quad\beta=\beta_{0}+\beta_{1}g_{v}.$
Together with the decomposition $\gamma=\alpha+\beta$ we also have the
estimate

$\displaystyle|\alpha|_{A}$ $\displaystyle=$
$\displaystyle|\alpha_{0}+\alpha_{1}g_{v}|_{A}\leq|\alpha_{0}|_{A}+|\alpha_{1}|_{A}|g_{v}|_{A}$
$\displaystyle\leq$ $\displaystyle
D_{1}\left(|\gamma_{0}|_{\overline{U_{C}}}+|\gamma_{1}|_{\overline{U_{C}}}|g_{v}|_{A}\right)$
$\displaystyle\leq$ $\displaystyle
D_{1}\left(|\gamma_{0}|_{\overline{U_{C}}}+|\gamma_{v}|_{\overline{U_{C}}}\frac{|g_{v}|_{\overline{U_{A}\cup
U_{B}}}}{m}\right)$ $\displaystyle\leq$ $\displaystyle
D_{1}(1+{M}/{m})|\gamma|_{\overline{U_{C}}}=D|\gamma|_{\overline{U_{C}}};$

the last inequality follows from the definition of the decomposition
$f=f_{0}+f_{v},$ since $f_{0}(z)=\frac{1}{2}(f(z)+\overline{f(\bar{z})})$ and
$f_{v}(z)=\frac{1}{2}(f(z)-\overline{f(\bar{z})})$ and so on an axially
symmetric compact set $K$ the estimates $|f_{0}|_{K},|f_{v}|_{K}\leq|f|_{K}$
hold. Similarly

$|\beta|_{B}\leq D_{1}(1+{M}/{m})|\gamma|_{\overline{U_{C}}};$

Proof for $\mathcal{SR}$. Let $\\{1,i,j,k\\}$ be the standard basis of
$\mathbb{H}$ and
$\gamma=\gamma_{0}+\gamma_{1}i+\gamma_{2}j+\gamma_{3}k\in\mathcal{S}\mathcal{R}(U)$
the unique decomposition of $\gamma$ with
$\gamma_{l}\in\mathcal{S}\mathcal{R}_{\mathbb{R}}(\Omega),l=0,\ldots,3.$ Since
for each $l$ we have
$|\gamma_{l}|_{\overline{U_{C}}}\leq|\gamma|_{\overline{U_{C}}}$, the case for
slice–preserving functions immediately implies that there exist
$\alpha_{l}\in\mathcal{SR}(U_{A}),$ $\beta_{l}\in\mathcal{SR}(U_{B}),$ such
that $\gamma_{l}=\alpha_{l}+\beta_{l}$ and the estimate
$|\alpha_{l}|_{A},|\beta_{l}|_{B}\leq D|\gamma_{l}|_{\overline{U_{C}}}$ holds
and hence $|\alpha|_{A},|\beta|_{B}\leq 4D|\gamma|_{\overline{U_{C}}}.$ ∎

###### Proof of Theorem 4.1 (1) and (3).

The proof of the claim (3) follows immediately once we have proved claim (1).

To prove claim (1), let an element in
$H^{1}(\Omega,\mathcal{SR}_{\mathbb{R}})$ be represented by a cocycle defined
on a sufficiently fine axially symmetric open covering $\mathcal{U}.$ Let
$\mathcal{B}=\\{U_{k},\,k\in\mathbb{N}_{0}\\}$ and
$\mathcal{B}^{1/(j+1)}=\\{U_{k}^{j}\\}_{k\in\mathbb{N}_{0}}$ be Cartan
coverings given by Theorem 1.1 in [PV2] subordinated to $\mathcal{U}$, where
the sets $\\{U_{k}^{j}\\}_{j\in\mathbb{N}_{0}}$ are a basis of open
neighbourhoods of $\overline{U_{k}}$. Let the cocycle be represented by the
functions $v_{kl}\in\mathcal{SR}(U^{0}_{kl})$. Our aim is to find functions
$v_{n}$ defined on $U_{n},$ $n\in\mathbb{N}_{0}$ so that $v_{kl}=v_{k}-v_{l}$
on $U_{kl}.$ Choose $\varepsilon>0$ and proceed by induction. of $U_{01}^{1}.$

$n=0.$ Define $A:=\overline{U_{0}^{2}},$ $B:=\overline{U_{1}^{2}}$,
$U_{A}:=U_{0}^{1}$, $U_{B}:=U_{1}^{1}$ and $U=U_{01}^{0}$. Then the Cartan
pair $(A,B)$ and sets $U_{A},U_{B},U$ fulfill the assumptions of Lemma 4.3.
Let $D$ be the constant given by this lemma. Because the set
$\overline{U_{01}^{1}}$ is a closed basic set, the function $c=v_{01}$ can be
approximated on $\overline{U_{01}^{1}}$ by an entire slice–regular slice
preserving function $h$, $|c-h|_{U_{01}^{1}}<\varepsilon/(2D)$. Lemma 4.3 (a)
for the function $c-h$ defined on $U_{01}^{0}$ yields slice–preserving
slice–regular functions $a_{1},b_{1}$ defined on $U_{A}$ and $U_{B}$
respectively, so that $c=a_{1}-b_{1},$ $|a_{1}|_{A}<\varepsilon/2$. Set
$v^{1}_{0}=a_{1}$ on $U_{0}^{1}$ and $v^{1}_{1}=b_{1}-h$ on $U_{1}^{1}.$ Then
$v_{01}:=v_{0}^{1}-v_{1}^{1}$ on a $U_{01}^{1}.$

$n\rightarrow n+1.$ Assume that the functions
$v^{n}_{k}\in\mathcal{SR}_{\mathbb{R}}(U^{n}_{k})$, $k=0,\ldots,n,$ have
already been defined. Set $A:=\overline{U_{0}^{n+2}\cup\ldots\cup
U_{n}^{n+2}}$ and $B=\overline{U_{n+1}^{n+2}}$,
$U_{A}=U_{0}^{n+1}\cup\ldots\cup U_{n}^{n+1}$, $U_{B}=U_{n+1}^{n+1}$ and
$U=(U_{0}^{n}\cup\ldots\cup U_{n}^{n})\cap U_{n+1}^{n}$. As above, the
requirements of Lemma 4.3 are fulfilled for sets $A,B,U_{A},U_{B}$ and $U$.
Let $D$ be the constant yielded by this lemma. Consider the functions
$v_{k\,n+1}$ with $k\leq n.$ The cocycle condition implies, that the functions
$v_{k\,n+1}-v_{k}^{n}$ defined on $U_{k\,n+1}^{n},$ $k\leq n,$ coincide on the
intersections of their domains of definition,

$v_{k\,n+1}-v_{k}^{n}=v_{l\,n+1}-v_{l}^{n}\Leftrightarrow
v_{k\,n+1}-v_{l\,n+1}=v_{k}^{n}-v_{l}^{n}=v_{kl},$

and hence define a function $c$ on $U$ By definition of Cartan pairs, the set
$\overline{U_{A}}\cap\overline{U_{B}}$ is a finite union of closed basic sets
therefore there exists an entire function $h$ so that $|c-h|_{U_{A}\cap
U_{B}}<\varepsilon/(D2^{n+1})$. By Lemma 4.3 (a), the function $c-h$ splits as
$c-h=a_{n+1}-b_{n+1}$ with functions $a_{n+1},b_{n+1}$ slice–regular
slice–preserving and defined on $U_{A}$ and $U_{B}$ respectively and the
estimate $|a_{n+1}|_{A}<\varepsilon/2^{n+1}$ holds. Set
$v_{k}^{n+1}:=v_{k}^{n}+a_{n+1}$ on $U_{k}^{n+1},$ $k\leq n$ and
$v_{n+1}^{n+1}:=b_{n+1}-h$ on $U_{n+1}^{n+1}.$ Obviously,
$v_{kl}=v_{k}^{n+1}-v_{l}^{n+1}$ holds for $k,l\leq n$. For $k\leq n$ we have

$v_{k}^{n+1}-v_{n+1}^{n+1}=v_{k}^{n}+a_{n+1}-b_{n+1}+h=v_{k}^{n}+c=v_{k\,n+1}\mbox{
on }U_{k\,n+1}^{n+1}$

hence $v_{kl}=v^{n+1}_{k}-v_{l}^{n+1}$ on $U_{kl}^{n+1}$ for all $k,l\leq
n+1$.

For each $k\in\mathbb{N}_{0}$ the sum $\sum_{n=k+1}^{\infty}a_{n}$ converges
uniformly on $\overline{U_{k}},$ so
$v_{k}:=v_{k}^{k}+\sum_{n=k+1}^{\infty}a_{n}=\lim_{n\rightarrow\infty}v_{k}^{n}$
is well defined and $v_{kl}=v_{k}-v_{l}$.

∎

In order to investigate the solvability of the second (or multiplicative)
Cousin problem, we start with the corresponding multiplicative Cartan’s
Splitting Lemmas for all the three types of classes of slice regular
functions.

######  Lemma 4.4 (Multiplicative Cartan’s splitting lemma with estimates).

Let $(A,B)$ be a Cartan pair, $C:=A\cap B$ and $U$ its open neighbourhood. Let
$U_{A}\Subset\mathbb{H}$ be a (topologically) regular axially symmetric open
neighbourhood of $A$, $U_{B}\Subset\mathbb{H}$ a (topologically) regular
axially symmetric open neighbourhood of $B$ so that
$(\overline{U_{A}},\overline{U_{B}})$ is a Cartan pair with $U_{C}:=U_{A}\cap
U_{B}\Subset U.$

Let $\varepsilon>0.$ For any $c\in\mathcal{SR}^{*}(U)$ there exist
$a\in\mathcal{SR}^{*}(U_{A})$ and $b\in\mathcal{SR}^{*}(U_{B})$ such that
$c=a*b$ on $U_{C}$ and $|a-1|_{A}<\varepsilon.$ In addition,

(a) if $c\in\mathcal{SR}^{*,+}_{\mathbb{R}}(U)$ then we can choose
$a\in\mathcal{SR}^{*,+}_{\mathbb{R}}(U_{A})$ and
$b\in\mathcal{SR}^{*,+}_{\mathbb{R}}(U_{B})$; if, moreover, $\overline{U_{C}}$
is a closed basic set, then for any $c\in\mathcal{SR}^{*}_{\mathbb{R}}(U)$ we
can choose $a\in\mathcal{SR}_{\mathbb{R}}^{*,+}(U_{A})$ and
$b\in\mathcal{SR}^{*}_{\mathbb{R}}(U_{B})$;

(b) if $\omega$ is a vectorial class on $\Omega\Supset(U\cup U_{A}\cup
U_{B})$, $g_{v}$ its global minimal representative on $\Omega$, $g_{v}\neq 0$
on $U$, and if $\gamma\in\mathcal{SR}_{\omega}(U),$ then we can choose
$\alpha\in\mathcal{SR}_{\omega}(U_{A})$ and
$\beta\in\mathcal{SR}_{\omega}(U_{B}).$

###### Proof.

Let ${U}^{\prime}_{A},{U}^{\prime}_{B}\Subset\mathbb{H}$ be two
(topologically) regular axially symmetric open neighbourhoods of $U_{A},U_{B}$
respectively, so that $(\overline{U^{\prime}_{A}},\overline{U^{\prime}_{B}})$
is a Cartan pair and
${U}^{\prime}_{C}:={U}^{\prime}_{A}\cap{U}^{\prime}_{B}\Subset U.$

Proof for $\mathcal{SR}_{\mathbb{R}}$. Because
$(\overline{U^{\prime}_{A}},\overline{U^{\prime}_{B}})$ is a Cartan pair, the
set $\overline{U^{\prime}_{C}}$ is a finite union of disjoint closed basic
sets. If it is a product set, then $c$ has a slice–preserving logarithm
$\gamma=\mathop{\text{missing}}{\rm log}\nolimits_{*}c$ on ${U}^{\prime}_{C}$
by [GPV, AdF2]. If $U^{\prime}_{C}\cap\mathbb{R}\neq\varnothing$ and since the
sign of $c|_{\mathbb{R}\cap U}$ is positive, i.e.
$c\in\mathcal{SR}^{+}_{\mathbb{R}}(U_{C}),$ then again it has a
slice–preserving logarithm $\gamma=\mathop{\text{missing}}{\rm
log}\nolimits_{*}c$ on ${U}^{\prime}_{C}$ by [GPV, AdF2]. Choose $\delta>0.$
By Runge Theorem 2.12 we can approximate $\gamma$ on the compact set
$\overline{U_{C}}$ by a slice–preserving slice–regular function
$\gamma^{\prime}$ on $\mathbb{H}$ as well as we want, in particular, we can
have $|\gamma-\gamma^{\prime}|_{\overline{U_{C}}}<\delta.$ By Lemma 4.3 (a),
there exist functions $\alpha^{\prime}\in\mathcal{SR}_{\mathbb{R}}(U_{A}),$
$\beta^{\prime}\in\mathcal{SR}_{\mathbb{R}}(U_{B})$ such that
$\gamma-\gamma^{\prime}=\alpha^{\prime}+\beta^{\prime}$ and the estimates
$|\alpha^{\prime}|_{{A}},|\beta^{\prime}|_{{B}}<D|\gamma-\gamma^{\prime}|_{\overline{{U}_{C}}}<\delta
D.$ Define $\alpha:=\alpha^{\prime}$ and
$\beta:=\beta^{\prime}+\gamma^{\prime},$ $a=\exp_{*}\alpha=\exp\circ\alpha,$
$b:=\exp_{*}\beta=\exp\circ\beta.$ Then $c=ab.$ The estimate
$|\alpha|_{{A}}<\delta D$ implies that for a small enough $\delta$ we have
$|a-1|_{U_{A}}<\varepsilon.$ It is clear that also
$a\in\mathcal{SR}_{\mathbb{R}}^{+}(U_{A}),$
$b\in\mathcal{SR}_{\mathbb{R}}^{+}(U_{B}).$

If $\overline{U_{C}}$ is a basic set then we may assume without loss of
generality that also $U^{\prime}_{C}$ is a basic set. If $U_{C}^{\prime}$ is a
product se, then we proceed as before. Assume that
$\mathbb{R}\cap{U_{C}^{\prime}}\neq\varnothing$. Then
$c|_{\mathbb{R}\cap{U_{C}^{\prime}}}$ has constant sign. If it is negative,
then $\gamma:=\mathop{\text{missing}}{\rm log}\nolimits_{*}(-c)$ exists and
$-c=ab$ for $a\in\mathcal{SR}_{\mathbb{R}}^{+}(U_{A}),$
$b\in\mathcal{SR}_{\mathbb{R}}^{+}(U_{B})$, hence we can write $c=a(-b).$

Proof for $\mathcal{SR}_{\omega}$. Let $\omega$ be a global vectorial class
and $g_{v}$ its minimal global representative. The conditions that
$U^{\prime}_{C}$ is a union of closed basic domains and $g_{v}\neq 0$ on $U$
ensure by [GPV, AdF2], that $\gamma:=\mathop{\text{missing}}{\rm
log}\nolimits_{*}c\in\mathcal{SR}_{\omega}(U_{C}^{\prime})$ exists and,
moreover, the set $\overline{U_{C}}$ is Runge in $\mathbb{H}$ (and in
$\Omega$). Decompose $\gamma=\gamma_{0}+\gamma_{1}g_{v}.$ We first approximate
$\gamma_{l}$ on $\overline{U_{C}}$ by
$\gamma_{l}^{\prime}\in\mathcal{SR}_{\mathbb{R}}(\mathbb{H})$, $l=0,1$, and
hence $\gamma$ by $\gamma^{\prime}\in\mathcal{SR}_{\omega}(\mathbb{H})$ and
then conclude as in the proof for $\mathcal{SR}_{\mathbb{R}}$, because for
functions in the same vectorial class the equality
$\exp_{*}(\alpha+\beta)=(\exp_{*}\alpha)*(\exp_{*}\beta)$ holds.

Proof for $\mathcal{SR}$. Because in general $\log_{*}c$ may not exist on
$U_{C}$ and the $*$-multiplication is not commutative in $\mathcal{SR}$, the
proof requires different tools and is formulated as Theorem 4.5 below. ∎

The additive Cartan’s Splitting lemma for slice–regular functions (Lemma 4.3)
together with the matrix representation of any $f\in\mathcal{SR}(\Omega)$
(with $\Omega$ an axially symmetric domain in $\mathbb{H}$) allows us to prove
the multiplicative Cartan’s Splitting lemma as in [GR]; the multiplicative
Cartan’s splitting lemma in [GR], VI.E, Theorem 7, used in one complex
variable and for the matrix functions $M_{f}$ on symmetric domains, boils down
to the following theorem because we have at our disposal the analogue of VI.E
Lemma 6 in [GR], i.e. the additive Cartan’s Splitting lemma 4.3.

######  Theorem 4.5.

Assumptions as in Lemma 4.4. Let $M_{c}\in\mathcal{GL}_{\mathcal{SR}}(U).$
Then there exist matrix functions
$M_{a}\in\mathcal{GL}_{\mathcal{SR}}(U_{A}),$
$M_{b}\in\mathcal{GL}_{\mathcal{SR}}(U_{B})$ such that $M_{a}M_{b}=M_{c}.$ In
addition, given $\varepsilon>0,$ the function $M_{a}$ can be chosen to satisfy

$|M_{a}-\mathbf{1}|_{A}<\varepsilon.$

######  Remark 4.6.

The original result in [GR], VI.E Theorem 7, is stated for the sets
$K,K^{\prime}$ and $K^{\prime\prime}$ which in one complex dimension are
rectangles. In our case we set
$K^{\prime}:=A_{I},K{{}^{\prime\prime}}:=B_{I},K:=C_{I}$, $F:=M_{c}|_{C_{I}}$
and the resulting matrix functions are $F^{\prime}:=M_{a}|_{A_{I}}$ and
$F^{\prime\prime}=M_{b}|_{B_{I}}$. Recall that the sets $A_{I}$ and $B_{I}$
have the separation property. Despite the fact that in our setting we do not
have such a nice geometric situation, the proof is adapted to our situation
more or less verbatim. In the sequel we focus only to parts of the proof where
we make some small changes in order to have the desired estimates.

The matrix norm $|\cdot|_{2}$ used in [GR] is induced by the Euclidean norm
and is of course equivalent to the matrix norm $|\cdot|$ we use, which is
induced by vector norm $|\cdot|_{1}$: for a $4\times 4$ matrix $M$ we have
$|M|/2\leq|M|_{2}\leq 2|M|$.

The proof starts by defining nested sequences $A_{n}=A_{I}+B(0,2^{-n}\delta),$
$B_{n}=B_{I}+B(0,2^{-n}\delta),$ for a small enough $\delta$ so that
$(\overline{A_{n}},\overline{B_{n}})$ form a Cartan pair; evidently the
lengths of the boundaries of $C_{n}=A_{n}\cap B_{n}$ are bounded from above by
some positive constant $L$, because the set $C_{I}=K$ has slice–piecewise
smooth boundary.

In the sequel we examine the estimates in [GR] for the matrix functions. The
proof is divided into two steps. First it is assumed that
$|F-\mathbf{1}|_{2,\overline{C_{1}}}<\rho/4$ for some small enough $\rho$
(fulfilling also some other conditions but this is not relevant for the
conclusions on the estimates of the product, once we know it converges). The
function $F^{\prime}$ is obtained as an infinite product
$F^{\prime}=\lim_{n\rightarrow\infty}(\mathbf{1}+G_{1}^{\prime})\cdot\ldots\cdot(\mathbf{1}+G^{\prime}_{n})$
with the estimates of the form $|G^{\prime}_{n}|_{K^{\prime}}<2^{-n}D\rho$ for
some positive constant $D,$ so that also $D\rho<1/2.$ Then we can estimate

$\displaystyle|F^{\prime}-\mathbf{1}|_{2,K^{\prime}}$ $\displaystyle\leq$
$\displaystyle e^{\sum_{n=1}^{\infty}|G^{\prime}_{n}|_{2}}-1\leq e^{\rho
2D}-1\leq\rho\ (2De).$

Hence, if $\varepsilon>0$ is given, one can choose $\rho$ to be so small, that
$4De\rho<\varepsilon,$ then we get the estimate $|M_{a}-\mathbf{1}|_{A}\leq
2|F^{\prime}-\mathbf{1}|_{2,K^{\prime}}<\varepsilon.$ A similar estimate holds
for $M_{b}$ on the set $B.$

If $|F-\mathbf{1}|_{2,\overline{C_{1}}}$ is not small enough, then by Runge
theorem (Theorem 2.13 rewritten in matrix representation form) we can
approximate $M_{c}^{-1}$ on $C_{1}$ by an invertible matrix function $M^{-1}$
defined on $\mathbb{H}$ so that
$|M_{c}^{-1}-M^{-1}|_{2,\overline{C_{1}}}<\rho/(4|M_{c}|_{2,\overline{C_{1}}})$.
Then $|M_{c}M^{-1}-\mathbf{1}|_{2,\overline{C_{1}}}<\rho/4.$ As a result we
get the decomposition $M_{c}M^{-1}=M_{a}M_{b}$ together with estimates on
$M_{a}-\mathbf{1},M_{b}-\mathbf{1}.$ The desired decomposition is then
$M_{c}=M_{a}M_{b}M$ with estimates on $M_{a}-\mathbf{1}$ still valid, while
the size of $M_{b}M-\mathbf{1}$ cannot be controlled. Similarly, if we
consider the decomposition $M^{-1}M_{c}=M_{a}M_{b}$ together with estimates on
$M_{a},M_{b}$, then $M_{c}=MM_{a}M_{b}$ with estimates on $M_{b}-\mathbf{1}$
still valid, while the size of $MM_{a}-\mathbf{1}$ cannot be controlled.

######  Remark 4.7.

In view of Remark 4.6, one immediately concludes that in the statement of
Lemma 4.4 the order of product and the roles of sets $A$ and $B$ can be
interchanged.

###  4.2. Solutions to the First and Second Cousin problem

As already mentioned, to prove Theorem 4.1 (2) we have to find a global
minimal representative for a given vectorial class. Let us first prove the
following lemma which is based on the fact that the Cartan covering has order
$2$ when restricted to the real axis.

######  Lemma 4.8.

Let $\Omega$ be an axially symmetric domain in $\mathbb{H}$. Given a cocycle
$v\in H^{1}(\Omega,\mathcal{SR}^{*}_{\mathbb{R}})$ there exists a Cartan
covering $\mathcal{U},$ the representative
$\\{v_{kl}\\}_{k,l\in\mathbb{N}_{0}}$ of $v$ and constants
$c_{l}\in\\{-1,1\\},l\in\mathbb{N}_{0}$, so that
$c_{k}^{-1}v_{kl}c_{l}\in\mathcal{SR}^{*,+}_{\mathbb{R}}(U_{kl}),k,l\in\mathbb{N}_{0}.$
Moreover,

$H^{1}(\Omega,\mathcal{SR}^{*}_{\mathbb{R}})\simeq
H^{1}(\Omega,\mathcal{SR}^{*,+}_{\mathbb{R}}).$

###### Proof.

First notice that every cocycle in $\mathcal{SR}^{*,+}_{\mathbb{R}}$ is also a
cocycle in $\mathcal{SR}^{*}_{\mathbb{R}}$. If two cocycles with values in
$\mathcal{SR}^{*,+}_{\mathbb{R}}$ are equivalent in
$\mathcal{SR}^{*,+}_{\mathbb{R}}$, then they are also equivalent in
$\mathcal{SR}^{*}_{\mathbb{R}}$. So we have a mapping
$H^{1}(\Omega,\mathcal{SR}^{*,+}_{\mathbb{R}})\rightarrow
H^{1}(\Omega,\mathcal{SR}^{*}_{\mathbb{R}})$.

We claim that we also have a mapping
$H^{1}(\Omega,\mathcal{SR}^{*}_{\mathbb{R}})\rightarrow
H^{1}(\Omega,\mathcal{SR}^{*,+}_{\mathbb{R}})$ and it is bijective. To see
this, choose a cocycle representing a given equivalence class with values in
$\mathcal{SR}^{*}_{\mathbb{R}}$. Every such class can be represented by
functions $v_{kl}\in\mathcal{SR}^{*}_{\mathbb{R}}(U_{kl})$ for a sufficiently
fine Cartan covering $\mathcal{U}=\\{U_{l}\\}_{l\in\mathbb{N}_{0}}$. If
$U_{kl}\cap\mathbb{R}=\varnothing$ then $v_{kl}\in\mathcal{SR}^{*,+}(U_{kl}).$
If $U_{kl}\cap\mathbb{R}\neq\varnothing$, then we try to find a different
representative of the cocycle. More precisely, we are looking for constants
$c_{m}\in\\{-1,1\\}$ so that $c_{k}^{-1}v_{kl}c_{l}$ is in
$\mathcal{SR}_{\mathbb{R}}^{*,+}$. We start by defining $c_{l}=1$ if
$U_{l}\cap\mathbb{R}=\varnothing.$

As already noted, the only $v_{kl}$-s which might not be in
$\mathcal{SR}^{*,+}(U_{kl})$ are the ones where
$U_{kl}\cap\mathbb{R}\neq\varnothing$. By construction of the Cartan covering,
each connected component of $\Omega\cap\mathbb{R}$ is covered by a chain of
topological open balls which do not intersect any other connected component of
$\Omega\cap\mathbb{R}$. We define the suitable constants $c_{l}$ on each
component separately.

Let $(x_{0},x_{1})\subset\mathbb{R}$ be a connected component of
$\Omega\cap\mathbb{R}$ and let $\mathcal{V}=\\{V_{l}\\}_{l\in\mathbb{Z}}$ be
the family of all the sets $U_{n}$ with
$U_{n}\cap(x_{0},x_{1})\neq\varnothing.$ These are topological open balls
which form a chain on $(x_{0},x_{1}),$ because by definition of a Cartan
covering, we have $U_{ijk}\cap\mathbb{R}=\varnothing$ for distinct $i,j,k.$
Fix one of these sets and denote it by $V_{0}.$ Then inductively denote by
$V_{l+1}$ the set which intersects $V_{l}$ on the right-hand side and by
$V_{l-1}$ the set which intersects $V_{l}$ on the left-hand side. For
$l\in\mathbb{Z},$ let $c_{l,l+1}=1$ if the sign of
$v_{l,l+1}|_{V_{l,l+1}\cap\mathbb{R}}$ is positive and $c_{l,l+1}=-1$
otherwise. Define $c_{0}=1$ and inductively define
$c_{l+1}=\Pi_{k=0}^{l}c_{k,k+1}$ for $l\geq 0.$ Similarly, for $l\leq 0,$ set
$c_{l}=\Pi_{k=l}^{0}c_{k,k+1}.$ Then $c_{l,l+1}=c_{l}^{-1}c_{l+1}.$ Replace
the representatives $v_{l,l+1}$ on the sets $V_{l,l+1}$ by
$c_{l}^{-1}v_{l,l+1}c_{l+1},$ $l\in\mathbb{N}_{0}.$ The new representatives
define the same cocycle and have values in $\mathcal{SR}_{\mathbb{R}}^{*,+}.$

Now assume that a given equivalence class $[v]$ is represented by the cocycle
$\\{v_{kl}\\}_{k,l\in\mathbb{N}_{0}}$ and also has representatives
$\\{u_{kl}\\}_{k,l\in\mathbb{N}_{0}}$ and
$\\{w_{kl}\\}_{k,l\in\mathbb{N}_{0}}$ with values in
$\mathcal{SR}_{\mathbb{R}}^{*,+}$. Then
$v_{kl}=c_{k}u_{kl}c_{l}^{-1}=d_{k}w_{kl}d_{l}^{-1}$ for some
$d_{k}\in\mathcal{SR}^{*}(U_{k}),k\in\mathbb{N}_{0}$. Hence
$u_{kl}=(c_{k}^{-1}d_{k})w_{kl}(c_{l}^{-1}d_{l})^{-1}.$ Because $w_{kl}$ and
$u_{kl}$ are in the class $\mathcal{SR}^{*,+}$ and the product commutes, also
$(c_{k}^{-1}d_{k})(c_{l}^{-1}d_{l})^{-1}$ is in $\mathcal{SR}^{*,+}(U_{kl})$.
If $U_{kl}$ intersects the real axis, it is a basic domain and so either both
$e_{k}:=c_{k}^{-1}d_{k}$ and $e_{l}:=c_{l}^{-1}d_{l}$ are in the class
$\mathcal{SR}^{*,+}$ or $-e_{k}$ and $-e_{l}$ are in the class
$\mathcal{SR}^{*,+}$. Consider the covering $\mathcal{V}$ containing the real
interval $(x_{0},x_{1})$ as above and assume that
$e_{0}\in\mathcal{SR}^{*,+}_{\mathbb{R}}(V_{0}).$ Then also $e_{-1}$ and
$e_{1}$ are in the class $\mathcal{SR}^{*,+}_{\mathbb{R}}$ and consequently
this holds for all sets that intersect $(x_{0},x_{1})$. If
$-e_{0}\in\mathcal{SR}^{*,+}_{\mathbb{R}}(V_{0})$ then replace $e_{k}$ by
$-e_{k}$ and again $u_{kl}$ and $w_{kl}$ are conjugated by functions from the
class $\mathcal{SR}^{*,+}_{\mathbb{R}}$.

So the mapping that maps the equivalence class of the cocycle
$\\{v_{kl}\\}_{k,l\in\mathbb{N}_{0}}$ with values in
$\mathcal{SR}_{\mathbb{R}}^{*}$ to the equivalence class of the cocycle
$\\{u_{kl}\\}_{k,l\in\mathbb{N}_{0}}$ with values in
$\mathcal{SR}_{\mathbb{R}}^{*,+}$ is well-defined.

Moreover, we have proved that if two cocycles are equivalent in
$\mathcal{SR}_{\mathbb{R}}^{*},$ they are equivalent in
$\mathcal{SR}_{\mathbb{R}}^{*,+}.$ Indeed, take two cocycles with values in
$\mathcal{SR}^{*}_{\mathbb{R}}$ and choose representatives of their
equivalence classes to be $\\{u_{kl}\\}_{k,l\in\mathbb{N}_{0}}$ and
$\\{v_{kl}\\}_{k,l\in\mathbb{N}_{0}}$ in $\mathcal{SR}^{*,+}_{\mathbb{R}}$
with respect to some Cartan covering $\\{U_{l}\\}_{l\in\mathbb{N}_{0}}$. Then
there exist $c_{l}\in\mathcal{SR}^{*}_{\mathbb{R}}(U_{l})$ so that
$v_{kl}=c_{k}^{-1}u_{kl}c_{l}$. Using the above notation consider
$\mathcal{V}$, the covering of the real interval $(x_{0},x_{1})$ and assume
without loss of generality that
$c_{0}\in\mathcal{SR}^{*,+}_{\mathbb{R}}(V_{0})$ (otherwise multiply all
$c_{l}$ by $-1$). But then also
$c_{l}=u_{l0}c_{0}v_{0l}\in\mathcal{SR}_{\mathbb{R}}^{*,+}(V_{0l})$ for
$l=1,-1$ and inductively, since the sets $V_{l}$ are basic, we have
$c_{l}\in\mathcal{SR}_{\mathbb{R}}^{*,+}(V_{l})$. This means that the mapping
$H^{1}(\Omega,\mathcal{SR}^{*}_{\mathbb{R}})\rightarrow
H^{1}(\Omega,\mathcal{SR}^{*,+}_{\mathbb{R}})$ is injective and surjective,
i.e. $H^{1}(\Omega,\mathcal{SR}^{*}_{\mathbb{R}})\simeq
H^{1}(\Omega,\mathcal{SR}^{*,+}_{\mathbb{R}}).$ ∎

######  Lemma 4.9.

The vectorial class $\omega$ on an axially symmetric domain $\Omega$ defines a
multiplicative cocycle in $H^{1}(\Omega,\mathcal{SR}_{\mathbb{R}}^{*,+}).$
More precisely, for every sufficiently fine Cartan covering
$\mathcal{U}=\\{U_{l}\\}_{l\in\mathbb{N}}$ there is a family of minimal local
representatives $\\{v_{l}\\}_{l\in\mathbb{N}}$ with
$v_{kl}:=v_{k}*v_{l}^{-*}\in\mathcal{SR}^{*,+}(U_{kl}).$

###### Proof.

Let $\mathcal{U}=\\{U_{l}\\}_{l\ \in\mathbb{N}}$ be a Cartan covering and for
each $l\in\mathbb{N}$ let $v_{l}$ be a minimal representative of $[v].$ By
definition of vectorial class, we have
$v_{kl}:=v_{k}*v_{l}^{-*}\in\mathcal{SR}_{\mathbb{R}}^{*}(U_{kl}),$ so it
defines a (multiplicative) cocycle in $\mathcal{SR}_{\mathbb{R}}^{*}(\Omega).$
By Lemma 4.8 there exist a representative of the same cocycle with functions
in $\mathcal{SR}^{*,+}_{\mathbb{R}}.$ ∎

Now we can prove Theorem 4.2 (1). For its solution we will make use of the
function $\log_{*}$; as expected this approach requires very accurate
description of the cocycles in the different settings considered.

###### Proof of Theorem 4.2 (1).

Fix an element in $H^{1}(\Omega,\mathcal{SR}_{\mathbb{R}}^{*,+}).$ Then there
exists a Cartan covering $\mathcal{U}=\\{U_{l}\\}_{l\in\mathbb{N}_{0}}$ of the
axially symmetric domain $\Omega$ and let
$\\{(v_{kl},U_{kl}),v_{kl}\in\mathcal{SR}_{\mathbb{R}}^{*,+}(U_{kl})\\}_{k,l\in\mathbb{N}}$
be the given multiplicative cocycle in $\mathcal{SR}_{\mathbb{R}}^{*,+},$
namely $v_{kl}\in\mathcal{SR}_{\mathbb{R}}^{*,+}(U_{kl})$,
$l,k\in\mathbb{N}_{0}$, are such that they satisfy conditions
$v_{kl}*v_{lk}=1$ on $U_{kl}$ and $v_{kl}*v_{lm}*v_{mk}=1$ on $U_{klm}$.

Thanks to [GPV] there exist slice–regular logarithms of $v_{kl}$, call them
$\mathop{\text{missing}}{\rm
log}\nolimits_{*}v_{kl}\in\mathcal{SR}_{\mathbb{R}}(U_{kl})$; the cocycle
conditions together with the fact that $U_{klm}$ is basic, imply that

(4.8) $\mathop{\text{missing}}{\rm
log}\nolimits_{*}v_{kl}+\mathop{\text{missing}}{\rm
log}\nolimits_{*}v_{lm}+\mathop{\text{missing}}{\rm
log}\nolimits_{*}v_{mk}=2\pi{\mathcal{I}}n_{klm}.$

This induces the cocycle
$C_{co}=\\{({\mathcal{I}}n_{klm},U_{klm})\\}_{k,l,m\in\mathbb{N}}$ which is
trivial by Theorem 3.9 i.e. there exists a cochain
$C=\\{(\mathcal{I}n_{kl},U_{kl})\\}_{k,l\in\mathbb{N}}$ of slice–preserving
functions which is mapped to $C_{co}$ by the coboundary operator.

Therefore functions $\mathop{\text{missing}}{\rm
log}\nolimits_{*}v_{kl}-2\pi\mathcal{I}n_{kl}$ form a cocycle of
slice–preserving functions. Since $H^{1}(\Omega,\mathcal{SR}_{\mathbb{R}})=0$
by Theorem 4.1, there exist slice–preserving functions
${v}_{l}^{\prime},l\in\mathbb{N},$ satisfying
$v_{kl}^{\prime}=v_{l}^{\prime}-v_{k}^{\prime}=\mathop{\text{missing}}{\rm
log}\nolimits_{*}v_{kl}-2\pi\mathcal{I}n_{kl},$ $k,l\in\mathbb{N}.$ Then
$v_{l}:=\exp_{*}(-v_{l}^{\prime})$ satisfy $v_{kl}=v_{k}v_{l}^{-1}.$ ∎

Lemma 4.9 and Theorem 4.2 (1) immediately imply the following.

######  Theorem 4.10.

Each globally defined vectorial class $\omega$ on an axially symmetric domain
$\Omega$ is principal, i.e. there exists
$g_{v}\in\mathcal{SR}_{\omega}(\Omega),$ so that $\omega=[g_{v}]$. Moreover,
$Z(\omega)=g_{v}^{-1}(0),$ with multiplicities, so $g$ is a global minimal
representative of $\omega.$

###### Proof.

As in the proof of Lemma 4.9, let $\mathcal{U}=\\{U_{l}\\}_{l\ \in\mathbb{N}}$
be a Cartan covering of $\Omega$, for each $l\in\mathbb{N}$ let $v_{l}$ be the
minimal representative of $[v]$ in $U_{l}$ and let, without loss of
generality,
$v_{kl}:=v_{k}^{-*}*v_{l}\in\mathcal{SR}_{\mathbb{R}}^{*,+}(U_{kl}),$ be the
induced cocycle.

Because $H^{1}(\Omega,\mathcal{SR}^{*,+}_{\mathbb{R}})=0,$
$v_{kl}=g_{k}^{-1}*g_{l}$ with $g_{l},g_{k}$ slice–preserving, so
$v_{l}*g_{l}^{-*}=v_{k}*g_{k}^{-*}$ so
$\\{(v_{l}*g_{l}^{-*},U_{l}),l\in\mathbb{N}\\}$ defines a function on
$\Omega.$ ∎

###### Proof of Theorem 4.1 (2).

Granted the existence of a global minimal representative of $\omega$, the
proof is the same as the proof of claim (1) in Theorem 4.1, with the
difference that the Cartan coverings have to be subordinated to $Z[\omega]$,
and we have to use Lemma 4.3 (b) in the place of Lemma 4.3 (a). ∎

Now we can also prove Runge Theorem 2.12 (2).

###### Proof of Theorem 2.12 (2).

Let $g_{v}$ be a global minimal representative and $K\Subset U.$ Let
${K}^{\prime}:=\widetilde{K}$ be the symmetrization of $K$ and
$\gamma=\gamma_{0}+\gamma_{1}g_{v}\in\mathcal{SR}_{\omega}(U)$ the
decomposition of $\gamma$. Since Runge theorem holds for slice–preserving
slice–regular functions, $\gamma_{l},l=0,1$ can be approximated on
${K}^{\prime}$ by slice–preserving functions $\tilde{\gamma}_{l}^{\prime}$ and
hence ${\gamma}^{\prime}:={\gamma}_{0}^{\prime}+{\gamma}_{1}^{\prime}g_{v}$
approximates $\gamma$ on $K.$ ∎

######  Remark 4.11.

For a class $\mathcal{F}$ of functions on a domain $\Omega$ which is closed
under multiplication, the second (or multiplicative) Cousin problem can be
stated in this version: given an open covering
$\mathcal{U}=\\{U_{\alpha}\\}_{\alpha}$ of a domain $\Omega$ and consider
functions $f_{\alpha\beta}\in\mathcal{F}(U_{\alpha\beta})$ which satisfy the
(multiplicative) cocycle or compatibility conditions

(4.9) $f_{\alpha\beta}\cdot f_{\beta\alpha}=1\,\,{\mathrm{in}}\ \
U_{\alpha\beta}\quad\mbox{ and }\quad f_{\alpha\beta}\cdot
f_{\beta\gamma}\cdot f_{\gamma\alpha}=1\,\,{\mathrm{in}}\ \
U_{\alpha\beta\gamma}$

for all $\alpha,\beta,\gamma$; it is required to find functions
$h_{\alpha}\in\mathcal{E}(U_{\alpha})$ and in the same class, such that
$f_{\alpha\beta}=h_{\alpha}^{-1}\cdot h_{\beta}$ for all $\alpha,\beta$.

In the noncommutative case, one would also look for the solution of the form
$f_{\alpha\beta}=h_{\beta}\cdot h_{\alpha}^{-1}$ for all $\alpha,\beta$. In
this case, the cocycle condition is of the form $f_{\gamma\alpha}\cdot
f_{\beta\gamma}\cdot f_{\alpha\beta}=1$.The order of writing the cocycle
condition does not affect the solvability of the second Cousin problem, since
they are linked by inverting the data, $f_{\alpha\beta}\rightarrow
f_{\beta\alpha}=f_{\alpha\beta}^{-1}.$ It rather changes the side where
factors $h_{\alpha}$ appear. If the cocycle is given by functions
$f_{\alpha},$ $f_{\alpha\beta}=f_{\alpha}^{-1}\cdot f_{\beta}$, then if
$f_{\alpha\beta}=f_{\alpha}^{-1}\cdot f_{\beta}=h_{\alpha}^{-1}\cdot
h_{\beta},$ the functions $\\{f_{\alpha}\cdot h_{\alpha}^{-1}\\}$ define a
global function. Similarly, if we have $f_{\alpha\beta}=f_{\beta}\cdot
f_{\alpha}^{-1}=h_{\beta}\cdot h_{\alpha}^{-1}$, then $\\{h_{\alpha}^{-1}\cdot
f_{\alpha}\\}$ define a global function on $\Omega$.

Hence, if $\mathcal{F}$ is one of the classes
$\mathcal{SR}_{\mathbb{R}},\mathcal{SR}_{\omega}$ or $\mathcal{SR}$ and we
want to preserve properties of the initial data such as zeroes and poles of
semiregular or regular functions, which are preserved under $*$-multiplication
from the right, then the first version has to be applied.

###### Proof of Theorem 4.2 (2), (3).

Let us first prove $(3)$. Let an element in $H^{1}(\Omega,\mathcal{SR}^{*})$
be represented by a cocycle defined on a sufficiently fine Cartan covering
$\mathcal{U}=\\{U^{\prime}_{k}\\}_{k\in\mathbb{N}_{0}}.$ Let
$\mathcal{B}=\\{U_{k}\\}_{k\in\mathbb{N}_{0}}$ and
$\mathcal{B}^{1/(j+1)}=\\{U_{k}^{j}\\}_{k\in\mathbb{N}_{0}}$ be Cartan
coverings given by Theorem 1.1 in [PV2]. Let the cocycle be represented by the
functions $v_{kl}\in\mathcal{SR}^{*}(U^{0}_{kl}),$ i.e the functions satisfy
the conditions

$v_{kl}*v_{lk}=1,\quad v_{klm}:=v_{mk}*v_{lm}*v_{kl}=1.$

To find nonvanishing functions $v_{k}$ defined on $U_{k},$
$k\in\mathbb{N}_{0}$ so that $v_{kl}=v_{l}*v_{k}^{-*}$ on $U_{kl}$, we follow
the scheme of the proof is the same as the one for Theorem 4.1 (1), with the
difference that we have to use multiplicative Cartan’s splitting lemma 4.4
with the decompositions of the form $c=b_{n}*a_{n}$ and estimates of the form
$|a_{n+1}-1|_{A}<\varepsilon/2^{n+1}$ in the place of additive Cartan’s
splitting lemma 4.3 and estimates $|a_{n+1}-0|_{A}<\varepsilon/2^{n+1}$. The
estimates then ensure that the product $\ldots a_{k+3}*a_{k+2}*a_{k+1}$
converges uniformly on $\overline{U_{k}}$.

For the proof of (2) and vectorial class $\omega$, let the coycle be
represented by functions $v_{kl}\in\mathcal{SR}^{*}_{\omega}(U_{kl})$ for a
sufficiently fine Cartan covering $\mathcal{U}$ subordinated to the pair
$(\\{\Omega\\},Z(\omega))$. By assumption, $U_{kl}\cap Z(\omega)=\varnothing$
for each $k\neq l$ and hence the Cartan’s Splitting lemma 4.4 (b) for
vectorial class applies. Then the proof goes as the case of slice–regular
functions with the only difference that the splitting is done within a given
vectorial class. ∎

######  Remark 4.12.

It is not possible to reduce the multiplicative Cousin problem in a given
vectorial class to the problem in $H^{2}(\Omega,\mathcal{I}\mathbb{Z}).$ Let
$g_{v}$ be the minimal representative of $\omega$ on $\Omega.$ Every cocycle
can be represented by functions $v_{kl}\in\mathcal{SR}^{*}_{\omega}(U_{kl})$
for some sufficiently fine Cartan covering $\mathcal{U}$ subordinated to
$(\\{\Omega\\},Z(\omega))$, so $U_{kl}\cap Z(\omega)=\varnothing.$ In [GPV]
(see also Proposition 3.5) it is shown that there exist logarithms
$\mathop{\text{missing}}{\rm
log}\nolimits_{*}v_{kl}\in\mathcal{SR}_{\omega}(U_{kl})$. The cocycle
condition then implies that

$\mathop{\text{missing}}{\rm
log}\nolimits_{*}v_{kl}+\mathop{\text{missing}}{\rm
log}\nolimits_{*}v_{lm}+\mathop{\text{missing}}{\rm
log}\nolimits_{*}v_{mk}=2\pi{\mathcal{I}}n_{klm}+2\pi\nu_{klm}\mathcal{J},\;\mathcal{J}:=\frac{g_{v}}{\sqrt{g_{v}^{s}}}.$

By Theorem 3.9 we know that the cocycle
$\\{{\mathcal{I}}n_{klm},k,l,m\in\mathbb{N}_{0}\\}$ is trivial. Unfortunately,
the imaginary unit function $\mathcal{J}$ is only locally defined and hence
this data do not induce (at least not in an obvious way) a cocycle problem
with integer coefficients. The problem is not that $g_{v}$ may vanish since we
have chosen the covering in such a way that $g_{v}$ does not have zeroes on
the sets $U_{kl}.$ The problem is that the square root $\sqrt{g_{v}^{s}}$ is
not necessarily globally defined since the domain which is the union of
$U_{kl}-s$ is not basic.

Next proposition is now straightforward:

######  Proposition 4.13.

Let $\Omega$ be an axially symmetric slice domain and $\omega=[g_{v}]$ a
vectorial class with vectorial function $g_{v}$ as its global minimal
representative. Assume that also $\sqrt{g_{v}^{s}}$ is globally defined.

If $Z(\omega)=\varnothing$, then
$\Gamma(\Omega,{\mathcal{K}}_{\omega})=\mathbb{Z}$.

If $Z(\omega)\neq\varnothing,$ then $\Gamma(\Omega,{\mathcal{K}}_{\omega})=0.$

###### Proof.

Notice that, because the domain $\Omega$ intersects the real axis, the periods
with $\mathcal{I}$ are excluded. If the vectorial class does not have a zero,
then a section from $\Gamma(\Omega,{\mathcal{K}}_{\omega})$ is locally equal
to $2n\pi\frac{g_{v}}{\sqrt{g_{v}^{s}}}$ for some $n\in\mathbb{Z}$ and the
identity principle implies that this holds globally.

If the vectorial class has a zero $z_{0}\in Z(\omega),$ then in a
neighbourhood of $z_{0}$, $\log_{*}1=2\pi n\mathcal{I}$ and hence, by identity
principle it is constant on every slice. Since the slices intersect the real
axis, $n=0.$ ∎

The following is a generalization of Theorem 4.1. Notice that the existence of
a globally defined minimal representative of $\omega$ is granted thanks to
Theorem 4.10.

######  Theorem 4.14.

Le $\Omega$ be an axially symmetric domain and let
$\mathcal{F}\in\\{\mathcal{SR}_{\mathbb{R}},\mathcal{SR}_{\omega},\mathcal{SR}\\}$.
Then $H^{q}(\Omega,\mathcal{F})=0$ every $q>0$.

###### Proof.

The triviality for $q=1$ was proved in Theorem 4.1. Triviality of higher
cohomology groups follows directly from the fact that a cocycle in
$\mathcal{F}$ can be represented on a sufficiently fine Cartan covering of
$\Omega$ and we have the corresponding additive splitting lemmas; in the case
of $\mathcal{F}=\mathcal{SR}_{\omega}$ the covering has to be subordinated to
$(\\{\Omega\\},Z[\omega])$. Any Cartan covering has order $3$, hence the
groups $H^{q}(\Omega,\mathcal{F})$ are trivial for $q\geq 3$.

Assume that $q=2$ and let an element in $H^{2}(\Omega,\mathcal{F})$ be
represented by a cocycle defined on a sufficiently fine Cartan covering
$\mathcal{U}=\\{U^{\prime}_{k}\\}_{k\in\mathbb{N}_{0}}$ of $\Omega$. Let
$\mathcal{B}=\\{U_{k}\\}_{k\in\mathbb{N}_{0}}$ and
$\mathcal{B}^{1/(j+1)}=\\{U_{k}^{j}\\}_{k\in\mathbb{N}_{0}}$ be the Cartan
coverings given by Theorem 1.1 in [PV2]. Let the cocycle on the Cartan
covering $\mathcal{B}^{1}=\\{U^{0}_{j}\\}_{j\in\mathbb{N}_{0}}$ be given by
functions $v^{0}_{klm}\in\mathcal{F}(U^{0}_{klm}),k,l,m\in\mathbb{N}_{0}$. The
claim follows from the fact that the decomposition of the form
$v_{klm}=v_{kl}+v_{lm}+v_{mk}$ can be done inductively, since there is no
cocycle condition to check. We briefly sketch the idea of the proof.

$n=1.$ Set $v^{0}_{01}=0$ and consider
$v^{0}_{012}\in\mathcal{F}(U^{0}_{012})$ if $U^{0}_{012}\neq\varnothing$.
Decompose $v^{0}_{012}-v^{0}_{01}$ defined on $U^{0}_{12}\cap U^{0}_{02}$
using the Cartan’s splitting lemma 4.3 to get
$v^{0}_{012}-v^{0}_{01}=v^{1}_{12}+v^{1}_{20}$ with $v^{1}_{12}$ defined on
$U^{1}_{12}$ and $v^{1}_{02}=-v_{20}^{1}$ defined on $U^{1}_{02}$. Set
$v_{01}^{1}:=v_{01}^{0}|_{U_{01}^{1}}$. If $U^{0}_{012}=\varnothing$, set
$v_{02}^{1}=v_{12}^{1}=0.$

$n-1\rightarrow n.$ Assume that $v^{n-1}_{kl}\in\mathcal{F}(U_{kl}^{n-1})$,
$l,k=0,\ldots n-1$, have already been constructed. By definition, the sequence
$\\{\overline{U^{n-1}_{0n}},\ldots,\overline{U^{n-1}_{n-1\,n}}\\}$ is a Cartan
string and hence the covering $\\{U^{n-1}_{0n},\ldots,U^{n-1}_{n-1\,n}\\}$ of
$U^{n}:=U^{n-1}_{0n}\cup\ldots\cup U^{n-1}_{n-1\,n}$ has order $2$ so no three
sets from the Cartan string intersect. In particular, the nerve of this
covering consists of a finite number of disjoint ’chains’, i.e. simplicial
complexes which are finite sequences of points and line segments.

Consider the set $U_{kln}^{n-1}$ for $k<l<n$ and assume that it is not empty.
We would like to show that the sets $\overline{U^{n-1}_{kn}}$ and
$\overline{U^{n-1}_{ln}}$ form a Cartan pair. Because the sequence
$\\{\overline{U^{n-1}_{0n}},\ldots,\overline{U^{n-1}_{ln}}\\}$ is a Cartan
string, the separation property is fulfilled for the sets
$A=\overline{U^{n-1}_{0n}}\cup\ldots\cup\overline{U^{n-1}_{l-1\,n}}$ and
$B=\overline{U^{n-1}_{ln}}$. Because $A\cap B=\overline{U_{kln}^{n-1}},$ the
separation property is fulfilled for the sets $\overline{U^{n-1}_{kn}}$ and
$\overline{U^{n-1}_{ln}}$ thus making them a Cartan pair.

The problem can be solved on each of the chains separately. Take one of the
above chains. If it consists only of a point, it corresponds to a set
$U_{kn}^{n-1}$ for some $k$ and we set $v_{kn}^{n}:=0$. Else, assume, for
simplicity, that the chain is defined by the sets
$U^{n-1}_{0n},\ldots,U^{n-1}_{kn}$ and proceed by induction. Choose a pair of
sets $(U^{1}_{A},U^{1}_{B})$ so that $U^{n-1}_{0n}\Supset U^{1}_{A}\Supset
U^{n}_{0n}$, $U^{n-1}_{1n}\Supset U^{1}_{B}\Supset U^{n}_{1n}$ and
$(\overline{U^{1}_{A}},\overline{U^{1}_{B}})$ is a Cartan pair. Then
$v^{n-1}_{01n}$ decomposes as $v^{n-1}_{01n}-v^{n-1}_{01}=v_{1n}-v_{0n}$ with
$v_{0n}\in\mathcal{F}(U^{1}_{A})$ and $v_{1n}\in\mathcal{F}(U^{1}_{B})$.
Assume that $v_{ln}$ have already been constructed for $l\leq m<k$ and let
$(U^{m}_{A},U^{m}_{B})$ be the last pair of sets. Choose a pair of sets
$(U^{m+1}_{A},U^{m+1}_{B})$ so that $U^{m}_{A}\cup U^{m}_{B}\Supset
U^{m+1}_{A}\Supset(U^{n}_{0n}\cup\ldots\cup U^{n}_{mn})$,
$U^{n-1}_{m+1\,n}\Supset U^{m+1}_{B}\Supset U^{n}_{m+1\,n}$ and
$(\overline{U^{m+1}_{A}},\overline{U^{m+1}_{B}})$ is a Cartan pair. Then
$v^{n}_{m\,m+1\,n}$ decomposes as
$v^{n}_{m\,m+1\,n}-v^{n-1}_{m\,m+1}-v_{nm}=v_{m+1\,n}-a$ with
$a\in\mathcal{F}(U^{m+1}_{A})$ and $v_{m+1\,n}\in\mathcal{F}(U^{m+1}_{B})$.
Put $v_{ln}^{\prime}:=v_{ln}-a$ for $l=0,\ldots,m$. Then
$v^{n}_{l_{1}l_{2}n}=v^{n}_{l_{1}l_{2}}+v_{l_{2}n}^{\prime}+v_{nl_{1}}^{\prime}=v^{n}_{l_{1}l_{2}}+v_{l_{2}n}-a+v_{nl_{1}}+a=v^{\prime}_{l_{1}l_{2}n}$.
Replace $v_{ln}$ by $v_{ln}^{\prime}$. When $m=k$ set
$v^{n}_{ln}:=v_{ln}|_{U_{ln}^{n}}$ for $l\leq k.$ This completes the induction
step and the proof.

∎

##  5\. Applications to divisors and jet interpolation theorems

In this section we present some consequences of the previously stated results.

######  Theorem 5.1 (Principal divisors).

Every divisor on an axially symmetric domain $\Omega\subset\mathbb{H}$ is
principal.

More precisely, let $\mathcal{B}=\\{U_{l}\\}_{l\in\mathbb{N}}$ and
$\mathcal{B}^{1}=\\{V_{l}\\}_{l\in\mathbb{N}}$ be Cartan coverings of $\Omega$
given by Theorem 1.1 in [PV2] and let the pairs
$\\{(V_{l},f_{l}),f_{l}\in\mathcal{SMR}(V_{l})\\}$ represent a given divisor.
Then there exists a global semiregular function $f\in\mathcal{SMR}(\Omega)$ so
that $f_{l}^{-*}*f\in\mathcal{SR}^{*}(U_{l}).$

###### Proof.

Let $Z$ be the support of the divisor. Without loss of generality we assume
that $\\{U_{j}\\}_{j\in\mathbb{N}}$ is a Cartan covering of $\Omega$
satisfying the condition $U_{kl}\cap Z=\varnothing$ for $k\neq n.$ By Theorem
1.1, there exist $h_{k}\in\mathcal{SR}^{*}$ so that
$f_{kl}=h_{k}*h_{l}^{-*}=f_{k}^{-*}*f_{l},$ hence $f_{l}*h_{l}=f_{k}*h_{k}$ so
$\\{f_{l}*h_{l}\in\mathcal{SMR}(U_{l}),l\in\mathbb{N}\\}$ defines a
semiregular function with prescribed zeroes and poles. ∎

To prove Main Theorem 2 (Theorem 1.2) we first need

######  Theorem 5.2 (Functions with prescribed zeroes).

Let $\Omega$ be an axially symmetric domain and $Z$ a discrete set of points
and spheres contained in $\Omega$. As in the Interpolation Theorem [PV1],
write $Z=\mathcal{S}\cup\mathcal{X}\cup\mathcal{Z},$ where

  1. (1)

$\mathcal{S}=\\{S_{j}=\\{a_{j}+Ib_{j},I\in\mathbb{S}\\},a_{j},r_{j}\in\mathbb{R},r_{j}>0,I\in\mathbb{N}\\}$
is the set of spheres

  2. (2)

$\mathcal{X}=\\{x_{l}\in\mathbb{R},l\in\mathbb{N}\\}$ is the set of real
points,

  3. (3)

$\mathcal{Z}=\\{z_{k}\in\mathbb{H}\setminus\mathbb{R},k\in\mathbb{N}\\}$ is
the set of nonreal points, so that
$\widetilde{\\{z_{k}\\}}\cap\mathcal{Z}=\\{z_{k}\\}$ for each
$k\in\mathbb{N}.$

Assume that to each sphere $S_{j}\in\mathcal{S}$ an integer $\sigma_{j},$ to
each $x_{j}\in\mathcal{X}$ an integer $\xi_{l}$ and to each
$z_{k}\in\mathcal{Z}$ an integer $\zeta_{j}$ is assigned. Then there exists
$f\in\mathcal{SR}(\Omega)$ with prescribed multiplicites at $Z;$ the
multiplicity of the sphere $S_{j}$ is $\sigma_{j},$ of the point $x_{l}$ is
$\xi_{l}$ and of the point $z_{j}$ is $\zeta_{j}$.

###### Proof.

By definition, each sphere $S_{j}$ contains at most one point from the set
$\mathcal{Z}.$ If it exists, we denote it by $z_{j}^{\prime}.$ Let
$\mathcal{P}=\\{P_{j},j\in\mathbb{N}\\}$ be the set of polynomials so that for
each $j\in\mathbb{N}$

  1. (1)

$P_{3j}$ has prescribed spherical multiplicities at the spheres $S_{j}$ and at
a point $z_{j}^{\prime}$ if it exists, and no other zeroes

  2. (2)

$P_{3j-2}$ has prescribed zeroes with multiplicities at the point $x_{j}$ and
no other zeroes and

  3. (3)

$P_{3j-1}$ has prescribed zeroes with multiplicities at the point
$z_{j}\not\in(\cup_{j}S_{j})\cap\mathcal{Z}$ and no other zeroes.

Define the spheres $\tilde{S}_{j}:=\widetilde{\\{z_{j}\\}}.$ Set
$\Omega^{\prime}=\Omega\setminus\widetilde{Z}.$ Then the sets
$U^{\prime}_{3j}:=\Omega^{\prime}\cup S_{j},$
$U^{\prime}_{3j-2}:=\Omega^{\prime}\cup\\{x_{j}\\}$ and
$U^{\prime}_{3j-1}:=\Omega^{\prime}\cup\tilde{S}_{j}$ define an axially
symmetric open covering $\mathcal{U}$ and $P_{k}^{-*}*P_{l}$ define a cocycle.
As customary, there exists finer Cartan coverings
$\mathcal{B}^{1/{(j+1})}=\\{U_{k}^{j},k\in\mathbb{N}_{0}\\}$, $j=0,1$,
subordinated to $(\mathcal{U},Z)$.

By construction, for each open set $U_{j}^{0}\in\mathcal{B}^{1}$ there is at
most one polynomial $P\in\mathcal{P}$ with zero in $U^{0}_{j}.$ If there is
such a $P$, define $v_{j}:=P^{-*}$ else define $v_{j}=1$ on $U^{0}_{j}.$

The functions $v_{j}$ define a cocycle $u_{kl}:=v_{l}*v_{k}^{-*}\in
U_{kl}^{0}$ of invertible functions.

The cocycle is trivial by Theorem 1.1 and hence there exist invertible
functions $h_{k}\in\mathcal{SR}^{*}(U_{k}^{1})$ so that
$v_{k}*v_{l}^{-*}=h_{k}*h_{l}^{-*}.$ Since *-multiplication from the right
hand side preserves the zeroes of the function on the left hand side, by
setting $u_{k}:=v_{k}^{-*}*h_{k}$ we define a slice–regular function on
$U_{k}^{1}$ with prescribed zeroes on $U_{k}^{1}$ and for each
$k,l\in\mathbb{N}_{0}$ the functions $u_{k}$ and $u_{l}$ coincide on
$U_{kl}^{1}$, hence the family
$\\{u_{k}\in\mathcal{SR}(U_{k}^{1}),k\in\mathbb{N}_{0}\\}$ defines a global
slice–regular function on $\Omega.$ ∎

We are now ready to proceed to the proof of Theorem 1.2.

###### Proof of Theorem 1.2.

We follow a similar approach as in the proof of Theorem 5.2. Let $\Omega$ and
$Z$ be as in Theorem 5.2. By definition, each sphere $S_{j}$ contains at most
one point $z_{j}^{\prime}$ from the set $\mathcal{Z}.$ Let
$\mathcal{P}=\\{P_{j},j\in\mathbb{N}\\}$ be the set of polynomials so that for
each $j\in\mathbb{N},$ the polynomial $P_{3j}$ has prescribed spherical jet
for the point $z_{j}^{\prime}$ if $S_{j}\cap\mathcal{Z}=\\{z_{j}^{\prime}\\}$
and a prescribed spherical jet on the sphere $S_{j}$ otherwise; similarly, let
polynomials $P_{3j-2},j\in\mathbb{N}$ have prescribed jets at the points
$x_{j}$ and $P_{3j-1},j\in\mathbb{N}$ at the points $z_{j}\neq
z_{k}^{\prime}$.

Let $g$ be a function with prescribed zeroes in $Z$ with multiplicities
strictly grater than the corresponding orders of jets as in Theorem 5.2.

Define the spheres $\tilde{S}_{j}:=\widetilde{\\{z_{j}\\}}.$ Then the sets
$U_{3j}:=\Omega\setminus S_{j},$
$U_{3j-2}:=\Omega\setminus\\{\tilde{x}_{j}\\}$ and
$U_{3j-1}:=\Omega\setminus\tilde{S}_{j}$ define an axially symmetric open
covering $\mathcal{U}$. As before, there exists finer Cartan coverings
$\mathcal{B}^{1/{j+1}}=\\{U_{k}^{j},k\in\mathbb{N}\\}$, $j\in\mathbb{N}_{0}$,
subordinated to $(\mathcal{U},Z)$

By construction, for each open set $U_{k}^{0}\in\mathcal{B}^{1}$ there is at
most one polynomial $P$ with prescribed jet in $U^{0}_{j}.$ If it exists,
define $Q_{j}=P$ else define $Q_{j}=0$.

The functions $v_{kl}:=g^{-*}*(Q_{k}-Q_{l})\in\mathcal{SR}(U_{kl}^{0})$ define
an additive cocycle of slice–regular functions which is trivial by Theorem 4.1
and hence there exists slice–regular functions
$v_{k}\in\mathcal{SR}(U_{k}^{1})$ so that $v_{kl}=v_{k}-v_{l}$. Then
$Q_{k}-Q_{l}=g*v_{k}-g*v_{l}\mbox{ on }U_{kl}^{1}.$ Since *-multiplication
from the right hand side preserves the zeroes of the function on the left hand
side, the set $\\{Q_{k}-g*v_{k},k\in\mathbb{N}\\}$ defines a global function
with prescribed jets. ∎

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