Differential Topological Aspects in Octonionic Monogenic Function Theory

In this paper we apply a homologous version of the Cauchy integral formula for octonionic monogenic functions to introduce for this class of functions the notion of multiplicity of zeroes and a-points in the sense of the topological mapping degree. As a big novelty we also address the case of zeroes lying on certain classes of compact zero varieties. This case has not even been studied in the associative Clifford analysis setting so far. We also prove an argument principle for octonionic monogenic functions for isolated zeroes and for non-isolated compact zero sets. In the isolated case we can use this tool to prove a generalized octonionic Rouché’s theorem by a homotopic argument. As an application we set up a generalized version of Hurwitz theorem which is also a novelty even for the Clifford analysis case.


Introduction
Especially during the last 3 years one notices a significant further boost of interest in octonionic analysis both from mathematicians and from theoretical physicists, see for instance [17][18][19]21,28]. In fact, many physicists currently believe that the octonions provide the adequate setting to describe the symmetries arising in a possible unified world theory combining the standard model of particle physics and aspects of supergravity. See also [22] for the references therein.
Already during the 1970s, but particularly in the first decade of this century, a lot of effort has been made to carry over fundamental tools from Clifford analysis to the non-associative octonionic setting.
This article is part of the Topical Collection on ISAAC 12 at Aveiro, July 29-August 2, 2019, edited by Swanhild Bernstein, Uwe Kaehler, Irene Sabadini, and Franciscus Sommen.
Many analogues of important theorems from Clifford analysis could also be established in the non-associative setting, such as for instance a Cauchy integral formula or Taylor and Laurent series representations involving direct analogues of the Fueter polynomials, see for example [16,[24][25][26][27]29]. Of course, one carefully has to put parenthesis in order to take care of the non-associative nature.
Although some of these fundamental theorems formally look very similar to those in the associative Clifford algebra setting(cf. [3]). Clifford analysis and octononic analysis are two different function theories.
In [18,19] the authors describe a number of substantial and structural differences between the set of Clifford monogenic functions from R 8 → Cl 8 ∼ = R 128 and the set of octonionic monogenic functions from O → O. This is not only reflected in the different mapping property, but also in the fact that unlike in the Clifford case, left octonionic monogenic functions do not form an octonionic right module anymore.
The fact that one cannot interchange the parenthesis arbitrarily in a product of octonionic expressions does not permit to carry over a number of standard arguments from the Clifford analysis setting to the octonionic setting.
In this paper we depart from the octonionic Cauchy integral formula for left or right octonionic monogenic functions, taking special care of the non-associativity by bracketing the terms together in a particular way. First we derive a topological generalized version of this Cauchy integral formula involving the winding number of 7-dimensional hypersurfaces in the sense of the Kronecker index. From the physical point of view this winding number represents the fourth Chern number of the G 2 -principal bundles that arise in the application of a generalization of 't Hoofd ansatz to construct special solutions of generalized G 2 -Yang-Mills gauge fields, see [4,13].
This homological version of Cauchy's integral formula is the starting point to introduce first the notion of the order of an isolated zero, or more generally, of an isolated a-point of a left (right) octonionic monogenic function. This notion of the order represents the topological mapping degree counting how often the image of a small sphere around zero (or around an arbitrary point a) wraps around zero (or a, respectively). An application of the transformation formula then leads to an explicit argument principle for isolated zeroes and a-points of octonionic monogenic functions. On the one-hand this argument principle naturally relates the fundamental solution of the octonionic Cauchy-Riemann equation with the fourth Chern number of the G 2principal bundles that are related to special solutions of the G 2 -Yang-Mills equation from 't Hoofd' ansatz. However, this topic will be investigated in detail in one of our follow-up papers.
On the other hand this argument principle allows us to establish a generalization of Rouché's theorem using a classical homotopy argument.
In turn, this version of Rouché's theorem enables us to prove that the limit function of a normally convergent sequence of octonionic monogenic functions that have no isolated a-points inside an octonionic domain either vanishes identically over the whole domain or it satisfies c∈C ord(f ; c) = 0. Note that this statement is slightly weaker than the classical Hurwitz theorem, because in the higher dimensional cases the condition ord(f ; c) = 0 does not immediately mean that f (c) = 0. It is a sufficient but not necessary condition for being zero-free. Anyway, this statement is also new for the associative Clifford analysis setting, of course one has to restrict oneself to paravector-valued functions when addressing this case.
A big goal and novelty of this paper consists in addressing also the context of non-isolated zeroes and a-points which lie on special simply-connected compact manifolds of dimension k ∈ {1, . . . , 6}. Instead of taking small spheres, the adequate geometric tool is the use of tubular domains that surround these zero or a-point varieties. This geometrical setting allows us to introduce the winding number of a surface wrapping around such a compact zero or a-point variety and gives a meaningful definition for the order of a zero variety of an octonionic monogenic function. We also manage to establish an argument principle for these classes of non-isolated zero varieties. These results are even new for the associative Clifford analysis setting and can also be applied to left and right monogenic paravector valued functions in R n+1 for general dimensions n ∈ N.
To finish we would like to mention that octonions also offer an alternative function theory of octonionic slice-regular functions, see for example [9,10,17]. There are of course also connections between octonionic sliceregular functions and octonionic solutions of the generalized octonionic Cauchy-Riemann equations. In the slice-regular context one even gets explicit relations between poles and zeroes as well as a simpler classification of zeroes in a very general situation. In the slice-regular setting only isolated and spherical zeroes can appear and their multiplicity can simply be d escribed in terms of a power exponent appearing in a factorization that makes use of the socalled slice-product. This is a very prosperous direction for developing further powerful function theoretical tools to address problems in the octonionic setting. Note that slice-regular functions also are connected with concrete physical applications, see for instance [4], in particular also in the construction of special solutions of 't Hoofd ansatz of G 2 -Yang-Mills solutions.
However, in this paper we entirely restrict ourselves to solutions of the octonionic Cauchy-Riemann equation, but it is an interesting challenge to pay more attention to these topics in the framework of other octonionic generalized function theories.

Basic Notions of Octonions
The octonions form an eight-dimensional real non-associative normed division algebra over the real numbers. They serve as a comfortable number system to describe the symmetries in recent unifying physical models connecting the standard model of particle physics and supergravity, see [4,11].
Following [2,7,31] and others, the octonions can be constructed by the usual Cayley-Dickson doubling process. The latter is initiated by taking two where · denotes the conjugation (anti-)automorphism which will be extended by (a, b) := (a, −b) to the set of pairs (a, b).
In the first step of this doubling procedure we get the real quaternions H. Each quaternion can be written in the form z = x 0 + x 1 e 1 + x 2 e 2 + x 3 e 3 where e 2 i = −1 for i = 1, 2, 3 and e 1 e 2 = e 3 , e 2 e 3 = e 1 , e 3 e 1 = e 2 and e i e j = −e j e i for all mutually distinct i, j from {1, 2, 3}. Already the commutativity has been lost in this first step of the doubling process. However, H is still associative.
The next duplification in which one considers pairs of quaternions already leads to the octonions O which are not even associative anymore. However, in contrast to Clifford algebras, the octonions still form a division algebra. In real coordinates octonions can be expressed in the form   Fortunately, the octonions still form an alternative and composition algebra.
In particular, the Moufang rule (ab)(ca) = a((bc)a) holds for all a, b, c ∈ O. In the special case c = 1, one obtains the flexibility condition (ab)a = a(ba).
Let a = a 0 + 7 i=1 a i e i be an octonion represented with the seven imaginary units as mentioned above. We call a 0 the real part of a and write a = a 0 . The conjugation leaves the real part invariant, but e j = −e j for all j = 1, . . . , 7. On two general octonions a, b ∈ O one has a · b = b · a.
The Euclidean norm and the Euclidean scalar product from R 8 naturally extends to the octonionic case by a, b := If z = 0 and r = 1 then we denote the unit ball and unit sphere by B 8 and S 7 , respectively. The notation ∂B 8 (z, r) means the same as S 7 (z, r).

Argument Principle for Isolated Zeroes of Octonionic Monogenic Functions
We start this section by recalling the definition of octonionic regularity or octonionic monogenicity in the sense of the Riemann approach. From [16,24] and elsewhere we quote It is clearly the lack of associativity that destroys the modular structure of O-regular functions. This is one significant structural difference to Clifford analysis. However, note that the composition with an arbitrary translation of the form z → z + ω where ω ∈ O still preserves monogenicity also in the octonionic case, i.e. Df (z + ω) = 0 if and only if Df (z) = 0. This is a simple consequence of the chain rule, because the differential remains invariant under an arbitrary octonionic translation.
An important property of left or right O-regular functions is that they satisfy the following Cauchy integral theorem, cf. for instance [26]: 7 and where n(z) is the outward directed unit normal field at z ∈ ∂G and dS = |dσ(z)| the ordinary scalar surface Lebesgue measure of the 7-dimensional boundary surface.
An important left and right O-regular function is the function q 0 : 8 whose only singular point is an isolated point singularity of order 7 at the origin. This function serves as Cauchy kernel in the following Cauchy integral formula for O-regular functions. Before we recall this formula, we point out another essential difference to the associative setting: Remark 3.3. As already mentioned in [13], in contrast to quaternionic and Clifford analysis, octonionic analysis does not offer an analogy of a general Borel-Pompeiu formula of the form ∂G g(z)dσ(z)f (z) = 0, not even if g is right O-regular and f left O-regular, independently how we bracket these terms together. The lack of such an identity is again a consequence of the lack of associativity. However, if one of these functions is the Cauchy kernel, then one obtains a generalization.

non-empty open set and G ⊆ U be an 8dimensional compact oriented manifold with a strongly Lipschitz boundary
Note that the way how the parenthesis are put is very important. Putting the parenthesis in the other way around, would lead in the left Oregular case to a different formula of the form where [a, b, c] := (ab)c − a(bc) stands for the associator of three octonionic elements. The volume integral which appears additionally always vanishes in algebras where one has the associativity, such as in Clifford algebras. See [26].
An important subcase is obtained when we take for f the constant function f (z) = 1 for all z ∈ U which is trivially left and right O-regular. Then the Cauchy integral simplifies to the constant value simply indicating if z belongs to the interior or to the exterior component of ∂G. This is the starting point to introduce a following generalized topological version of the above stated Cauchy integral formula. Following for instance [14] one can consider more generally for G a bounded Lipschitz domain whose boundary ∂G could be a 7-chain, homologous to a differentiable 7-chain with image ∂B(z, r), parametrized as In this more general case one has where w ∂G (z) represents the topological winding number, sometimes called the Kronecker-index (cf. [12]), counting how often ∂G wraps around z. Note that this is a purely topological entity induced by where H 8 is the related homology group andH 7 the related reduced homology group. Due to this property, the winding number w ∂G (z) is always an integer. This is the basis for the more general topological version of Cauchy's integral formula: is the topological winding number counting how often Γ wraps around z. The latter equals zero if z in a point from the exterior of G.
Remark 3.6. Note that if we put the parenthesis the other way around, then we get the identity 3 The volume integral is not affected in the topological version, because we simply integrate over the volume and orientation does not play any role, because the scalar differential dV = dw 0 ∧ · · · ∧ dw 7 has no orientation.
Remark 3.7. Comparing with [4,13], we can relate the octonionic winding number with the fourth Chern number of the G 2 -principal bundles associated to special solutions of G 2 Yang-Mills gauge fields arising in generalizing 't Hoofd ansatz, see [4,13]. This allows us to explicitly relate the fundamental solution of the octonionic Cauchy-Riemann equation with Chern numbers of the related G 2 -principal bundles. We will shed some more light on this interesting connection in a follow-up paper.
The topological winding number is also the key tool to define a generalized notion of multiplicity of zeroes and a-points of O-regular functions. To proceed to the definition and classification of a-points we first need the octonionic identity theorem: Although the proof only uses basic tools of octonionic analysis, we prefer to present it in detail, as we are not aware of a direct reference in the literature addressing the particular octonionic setting. For the proof of the statement in the associative Clifford analysis setting we refer to [12], p. 187.
Proof. The proof can be done by extending Fueter's argumentation from the quaternionic case presented in [8] on pp.185-189. Without loss of generality we consider the situation where g(z) is the zero function. Suppose now that V is a seven dimensional smooth manifold where f | V = 0. Consider an arbitrary point c ∈ V with f (c) = 0. Since V is 7-dimensional and smooth one can find seven R-linearly independent unit octonions, say n 1 , . . . , n 7 with |n h | = 1 (h = 1, . . . , 7) that lie in the 7-dimensional tangent space T V (c). Next define ξ are real for all j = 0, 1 . . . , 7 and all h = 1, . . . , 7. Next consider for each point c ∈ V the real 7 × 8-matrix composed by the seven rows constituted by the eight real coordinates of the seven octonions n 1 , . . . , n 7 , respectively, i.e.
Re-interpreting the seven octonions n j as column vectors from R 8 , we have rank(n 1 , . . . , n 7 ) = 7 in view of the R-linear independency. Consequently, also the rank of the largest non-vanishing sub-determinant must equal 7.
Without loss of generality we may suppose that 1 ξ 2 · · · ξ Otherwise, we change the labels of the components.
Next we use that f (z) = f 0 (z) + 7 k=0 f k (z)e k ≡ 0 on V . Therefore, the directional derivatives also vanish all, i.e. ∂f ∂n h = 0 for each h = 1, 2, . . . , 7. Using the ordinary chain rule gives seven equations: Additionally, as eighth condition, f has to satisfy the octonionic left Cauchy-Riemann equation 0 ξ 1 · · · ξ at each z ∈ V . Note that also the octonionic Cauchy integral formula implies that the left O-regularity of f is also inherited by all partial derivatives of f . Consequently, the same argumentation is also true for all partial derivatives ∂ n 1 +···+n 7 ∂x n 1 1 ···∂x n 7 7 f (z) = 0. Following [16,25] we can expand f into a Taylor series around each left O-regular point z = c ∈ V of the form One has to apply the parenthesis in this particular way. Due to the lack of associativity, the parenthesis cannot be neglected. Here, perm(n) denotes the set of all distinguishable permutations of the sequence (n 1 , n 2 , . . . , n 7 ) and   [10,17], then one even gets a much stronger version of the identity theorem, namely stating that two slice-regular functions already coincide with each other, when they coincide with each other on a onedimensional set with an accumulation point. This has a strong consequence on the structure of the zeroes.
Since also the octonions form a normed algebra, we can introduce the notion of an isolated a-point of an O-regular function as follows, compare with [14,20]: = 0. However, this clearly is just a sufficient criterion, as the following example illustrates. Take for instance the function : O → O defined by f (z) := V 2,0,0,...,0 (z) + V 0,2,0,...,0 (z) + · · · + V 0,...,0,2 (z) which is clearly left and right O-regular in the whole algebra O. Obviously, one has f (0) = 0. In general, f (z) = 0 implies that first x 2 1 + x 2 2 + · · · + x 2 7 = 7x 2 0 , and one has x 0 x i = 0 for each i = 1, . . . , 7. The first relation implies that Inserting this expression into the other relations yields x 2 1 + · · · + x 2 7 x i = 0 for all i = 1, . . . , 7. Since x 2 1 + · · · + x 2 7 > 0 whenever (x 1 , x 2 , . . . , x 7 ) = (0, 0, . . . , 0) we must have x i = 0 for all i = 1, . . . , 7. Therefore, also x 0 = 0. Summarizing z = 0 is the only zero of f and therefore it must be an isolated zero. The Jacobian matrix however is: Inserting z = 0 yields det(Jf)(z) = 0. A typical example of a non-linear left O-regular function with one single octonionic isolated zero z * satisfying Jf(z * ) = 0 can be constructed by applying Hempfling's construction from [14] p. 111. Adapting from [14], the octonionic version of the function actually is left O-regular. We have So, f actually satisfies ∂f ∂x0 + 7 k=1 e k ∂f ∂x k = 0. As one readily observes, one has f (z * ) = 0 when inserting z * = 1 + e 1 + · · · + e 7 . Furthermore, ∂fi ∂xj = δ ij 7 k=0,k =i,j x k , where δ ij denotes the and therefore det(Jf((1 + e 1 + · · · + e 7 ))) = −7 = 0. z * clearly is an isolated zero of f . Note that in general a left or right O-regular function can possess also zeroes that lie on k-dimensional manifolds with k ≤ 6. The case k = 7 cannot appear as direct a consequence of Proposition 3.8, because if a left O-regular function vanishes on a 7-dimensional manifold, then it must be identically zero over the whole 8-dimensional space. Furthermore, note that the zero sets of left or right O-regular functions must be real analytic manifolds. Already very simple octonionic functions can have connected sets of zeroes. Adapting from [31] and [14], in the octonionic case the simplest examples (for each dimension) are where Z i are again the octonionic Fueter polynomials Generalizing the construction from [14] a further class of interesting examples can be gained from the following construction. Let k ∈ {2, . . . , 6} be an integer and consider the function f : Z j e j composed by the octonionic Fueter polynomials. Again, this function is both left and right O-regular and can be written in the form only if the following system of equations is satisfied − R k , provided the value in the square root expression is not negative. In this case the zero set consists at most of two isolated points (x 0 , 0, . . . , 0) on the real axis.
In the spirit of [14,15,20] we now proceed to introduce the order of an isolated zero or isolated a-point of an O-regular function. This can be done like in the quaternionic and Clifford analysis case in terms of the topological Cauchy integral mentioned above and then represents the order of an isolated a-point in the sense of the topological mapping degree.
So, in the case where h(z) = 1 for all z ∈ U , one has In view of the mentioned property H 8 (Γ, Γ − c) ∼ =H7(S7) one can replace in the latter equation Γ by the homeomorphic equivalent small sphere ∂B(c, ε), so we have Next we replace the octonion y by f (c) − a and ∂B(c, ε) by (f − a)(∂B(c, ε)) and one obtains We recall that also f (∂B(c, ε)) and hence also the translated expression (f − a)(∂B(c, ε)) represents a 7-dimensional cycle, cf. [1] p. 470.  Since the situation is already so complicated in the quaternions, it cannot be expected that one gets a simpler relation for the octonionic case. Actually, analogues of the counter-example presented in [31] on p. 131-132 can easily be constructed.
In the octonionic slice-regular setting described for instance in [9], the situation is much simpler. As mentioned previously, in the slice function theoretical setting an octonionic slice-regular function either has isolated zeroes or spherical zeroes, similarly to the slice-monogenic setting in R n+1 cf. [5,9]. In terms of the symmetric slice product the multiplicity of such a zero then can be described by the exponent of the (slice) power, namely in the usual way like in classical real and complex analysis: A slice-regular function f can be decomposed uniquely in the way f (z) = (z − a) * k * g(z) where g(z) is a uniquely defined and zero-free slice-regular function around a, see [5,9] and elsewhere. Note that ordinary powers of z are intrinsic slice regular functions, also in the octonions. The slice-product gives some kind of symmetric structure. In the setting of O-regular functions in the sense of the Cauchy-Riemann operator, such a decomposition is not possible, because of the lack of commutativity (and also of non-associativity).
The definition of the order of an isolated a-point of an octonionic left or right O-regular function in the sense of Definition 3.11 is very natural from the topological point of view and so far the only meaningful tool to introduce a notion of "multiplicity" of an a-point. However, using this definition to calculate the value of the order of a concrete practical example is very difficult in general. Note that one has to perform the integration over the image of the sphere. Now, a significant advantage of the octonionic setting in comparison to the Clifford analysis setting is that octonionic functions represent maps from O → O which can be uniquely identified with a map from R 8 → R 8 , by identifying the map x 0 + x 1 e 1 + · · · + x 7 e 7 → f 0 (z) + f 1 (z)e 1 + · · · + f 7 (z)e 7 with the corresponding map . . .
Clifford analysis one deals with maps from R 8 to Cl 8 ∼ = R 128 . Now, if the 7-dimensional surface ∂G is parametrized as in (1), the image of that surface f (∂G) can be parametrized as and one can simply apply the chain rule for ordinary real differentiable functions from R 8 → R 8 , as indicated in [14] for purely paravector-valued functions. Applying the chain rule and exploiting the special mapping property that the image of octonionic functions are again octonions leads to the following octonionic generalization of the transformation formula from [20] p. 32. In the Clifford analysis case one had to restrict onself to particular paravectorvalued functions. This restriction is not necessary in the octonionic setting: It should be pointed out very clearly that does not mean the usual octonionic product. To be more explicit [dσ(z)] is interpreted as the vector The adjunct matrix [(Jf) adj (z)] has the form This also provides a correction to [20] p. 32 where the index i of the function f has been forgotten as well as the star after (Jf) (indicating the adjunct) in the second line of the proof. The proof for the octonionic case can be done along the same lines as presented for the paravector-valued Clifford case in [20] p. 32. The chain rule leads to and the stated formula follows, because no associativity property is required. This lemma allows us to reformulate the definition of the order given in Definition 3.11 in the way that the integration is performed over the simple sphere S 7 (c, ε). In contrast to the Clifford analysis case presented in [20] p. 33 we do not need to worry about a possible restriction of the range of values. All octonion-valued functions satisfying the left or right octonionic Cauchy-Riemann system are admitted here. However, the way how we put the brackets in the following theorem is crucially important. In the left Oregular case we have Then the order of the a-point can be re-expressed by Here, · stands for the octonionic product, where the term inside the large parenthesis on the right is re-interpreted as octonion.
Note that the Jacobian determinant is invariant under translations. Therefore J(f − a)(z) = Jf(z). In the complex case the Jacobian simplifies to (f − a) (z) = f (z) and one re-obtains the usual integrand f (z) f (z)−a because the Cauchy kernel then coincides with the simple inverse.
For the sake of completeness, in the right O-regular case one obtains Note that we always have ord(f −a; c) = 0 in all points c where f (c) = a. As a direct application this property and the statement of Theorem 3.16 we can deduce the following argument principle for isolated a-points of O-regular functions which provides an extension of Theorem 1.34 from [20] where the paravector-valued Clifford holomorphic case has been treated. But also in the octonionic case we have Proof. The proof follows along the same lines as in the Clifford analysis case given in [20] p. 33. This is a consequence of its predominant topological nature. The crucial point is that any oriented compact manifold can have atmost finitely many isolated a-points in its interior, let us call them c 1 , . . . , c n . Thus, one can find a sufficiently small real number ε > 0 such that there are no a-points in the union of the sets n i=1 B(c i , ε)\{c i }. Since f has neither further a-points nor singular points in the remaining part C\ n i=1 B i one obtains in view of Theorem 3.16 that The assertion now follows directly, when we take into account the mentioned property that ord(f − a; c) = 0 at all c ∈ C with f (c) = a.
The big goal of the argument principle is that it provides us with a topological tool to control the isolated a-points or zeroes of an octonionic regular function under special circumstances. Its classical application is Rouché's theorem that presents a sufficient criterion to describe by which function an octonionic regular function may be distorted in the way that it has no influence on the numbers of isolated zeroes inside a domain, when particular requirements are met. Alternatively, it gives a criterion to decide whether two octonionic monogenic functions have the same number of isolated zeroes inside such a domain. In close analogy to the associative Clifford analysis case, cf. [20] Theorem 1.35, we may establish Also the nature of this theorem is predominantly topological. The topological aspects play a much more profound role than the function theoretical aspects, which nevertheless are also needed because the proof uses the argument principle involving the particular Cauchy-kernel of the octonionic Cauchy-Riemann system. Let us define a family of left O-regular functions depending on a continuous real parameter t ∈ [0, 1] by For each t ∈ [0, 1] each function h z is left O-regular over G, since t is only a real parameter. Note that otherwise, the left O-regularity would be destroyed in general. Let z ∈ Γ. Then we have |t(g(z) − f (z)| = |t||f (z) − g(z)| ≤ |f (z)−g(z)| < |f (z)|, where the latter inequality follows from the assumption. Therefore h t (z) = 0 for all z ∈ Γ.
Furthermore, for each t ∈ [0, 1] the entity ord(h t ; c) is an integer. Since the number of zeroes is supposed to be finite in G, for each t the sum 51 c∈C ord(h t ; c) is finite and represents a finite integer N (t) ∈ Z. Per definition we have Since all terms under the latter integral are continuous functions in the variable t, also the expression N (t) on the left-hand side must be continuous in the variable t. However, N (t) is an integer-valued expression for any t ∈ [0, 1]. Therefore, N (t) must be a constant expression, hence N (0) = c∈C ord(h 0 ; c) = c∈C ord(f ; c) and N (1) = c∈C ord(h 1 ; c) = c∈C ord(g; c) must be equal.
As a nice application of Theorem 3.18 we can establish the following weakened version of Hurwitz' theorem. The following statement can also be carried over to the quaternionic monogenic setting and to the context of paravector-valued monogenic functions in Clifford algebras, for which this statement has not been established so far, at least as far as we know. We prove Proof. According to [26] Theorem 11, left (or right) O-regular functions satisfy Weierstraß' convergence theorem. Therefore, the limit function f is a well-defined O-regular function over the whole domain G. Let us assume now that f ≡ 0 over G. Take an arbitrary point z * ∈ G. In view of the identity theorem of left O-regular functions (Proposition 3.8) there must exist a positive real R > 0 such that the closed ball B(z * , R) is entirely contained inside G and M := min z∈S7(z * ,R) |f (z)| > 0. Moreover, since S 7 (z * , R) is compact there must exist an index n 0 ∈ N such that Working on the intersection of algebraic geometry and hypercomplex function theories represents a promising branch for future investigation. Furthermore, this paper shows that the argument principle is more a topological theorem than an analytic one, although the Cauchy kernel is explicitly needed in its definition. However, the predominant topological character gives the hope that these kinds of theorems can be carried over to many more hypercomplex function theories, in particular to the context of null-solutions to other differential equations. However, a really substantial question is to ask whether these tools can be carried over to functions that are defined in other algebras beyond octonions and paravector-valued subspaces of Clifford algebras. Both paravector spaces and octonions are normed and are free of zero-divisors. Following Imaeda [16], already in the context of sedenions it is not anymore possible to set up a direct analogue of Cauchy's integral formula. Cauchy's integral formula however is the basic tool for establishing all these results. The appearance of zero divisors will also have an impact on the topological properties. There remain a lot of open questions and challenges for future research.
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