arxiv: 2604.12876 · v1 · submitted 2026-04-14 · 🧮 math.CV · math.RA

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Fueter trees for Dunkl-regular functions over alternative *-algebras

Alessandro Perotti

Pith reviewed 2026-05-10 13:51 UTC · model grok-4.3

classification 🧮 math.CV math.RA

keywords Fueter theoremDunkl-regular functionsalternative *-algebrasFueter treesiterated Laplacianhypercomplex spacespartitions into odd partsDunkl-monogenic functions

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The pith

Fueter theorems over alternative *-algebras correspond one-to-one with graphs called Fueter trees, whose count in dimension n+1 equals the number of odd-part partitions of n.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a general Fueter theorem for Dunkl-regular functions on real alternative star-algebras. It shows that suitable powers of the Laplacian send these functions to Dunkl-monogenic functions that possess axial symmetry. By embedding known hypercomplex function theories inside the Dunkl-monogenic class, the result unifies earlier Fueter-type statements and gives the most general description of how the Laplacian acts on such function spaces. The central device is a bijection between Fueter theorems and Fueter trees, graphs that record the successive mappings produced by the iterated Laplacian. This bijection immediately yields the count of distinct trees as the number of partitions of n into odd parts.

Core claim

Fueter theorems are in one-to-one correspondence with Fueter trees that describe the interactions between Dunkl-regular function spaces and the iterated Laplacian; the number of distinct Fueter trees on a hypercomplex space of dimension n+1 equals the number of partitions of the integer n into odd parts.

What carries the argument

Fueter trees, graphs whose vertices and edges encode the distinct ways the iterated Laplacian maps spaces of Dunkl-regular functions into spaces of Dunkl-monogenic functions with axial symmetry.

If this is right

Several previously separate Fueter-type results for Clifford analysis, quaternionic analysis and other hypercomplex settings become special cases of the single statement.
The action of any power of the Laplacian on function spaces over a hypercomplex subspace of dimension n+1 is completely classified by the corresponding Fueter tree.
Each Fueter tree gives an explicit recipe for constructing the image of a Dunkl-regular function under the appropriate Laplacian power.
The combinatorial count implies that exactly as many independent Fueter-type theorems exist as there are partitions of n into odd parts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same tree enumeration might classify analogous statements for other families of monogenic or regular functions that admit a Laplacian mapping.
Explicit construction of the functions attached to each tree in low dimensions could produce new closed-form solutions not yet recorded in the literature.
The appearance of odd-part partitions suggests a hidden link to generating functions or symmetry groups that might be exploited to prove further identities.

Load-bearing premise

The embedding of hypercomplex function theories into the larger class of Dunkl-monogenic functions preserves all algebraic and analytic properties needed for the Laplacian mappings and the tree correspondence to hold without extra constraints.

What would settle it

Exhibit one concrete alternative *-algebra and dimension n+1 in which the number of distinct Laplacian-induced mappings from Dunkl-regular functions to axially symmetric Dunkl-monogenic functions differs from the number of odd-part partitions of n.

Figures

Figures reproduced from arXiv: 2604.12876 by Alessandro Perotti.

**Figure 2.** Figure 2: The Fueter tree for Clifford slice-monogenic functions is a unary tree of height |𝜅| = (𝑛 − 1)/2. The tree has root SM (Ω), internal nodes some spaces FP (Ω) (generalized partial-slice monogenic functions of type (2𝑖, 𝑛−2𝑖), 𝑖 = 1, . . . , (𝑛 − 3)/2) and one leaf M (Ω). O = H + ℓH. The Dunkl-Cauchy-Riemann operator with multiplicities k = − 1 3 (1, 1, 1, 0, 1, 1, 1) is 𝐷P = 𝜕O − 1 3 ∑︁ 7 𝑖=1,𝑖≠4 𝑣𝑖 𝑥𝑖 (1 −… view at source ↗

**Figure 4.** Figure 4: Binary Fueter tree for octonionic Dunklregular functions with root FP (Ω), where P = {{1, 2, 3}, {4}, {5, 6, 7}}. There are two nodes FP1 (Ω) and FP1 (Ω) with P1 = {{1}, . . . , {4}, {5, 6, 7}}, P3 = {{1, 2, 3}, {4}, . . . , {7}} and two leaves M (Ω). computation shows that 𝑇1 ( 𝑓 ) = −𝛿 1 2 𝑓 = − 4 3 𝑥 3 0 + 4𝑥0 [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗

read the original abstract

We prove a general Fueter Theorem over real alternative *-algebras. We show that a suitable power of the Laplacian maps Dunkl-regular functions to Dunkl monogenic functions with axial symmetries. Using the embedding of hypercomplex function theories in the class of Dunkl monogenic functions, we subsume several Fueter-type results known in the literature and obtain the most general form for the action of the Laplacian on function spaces over hypercomplex subspaces. We show that Fueter Theorems are in a one-to-one correspondence with a class of graphs, the Fueter trees, that describe the interactions between Dunkl-regular function spaces and the relation with the iterated Laplacian. We obtain that the number of distinct Fueter trees on a hypercomplex space of dimension $n+1$ is equal to the number of partitions in odd parts of the integer $n$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Fueter trees classify Laplacian mappings for Dunkl-regular functions via a bijection with known theorems, and the count equals odd-part partitions of n, but this rests on the embedding preserving exact Laplacian behavior.

read the letter

The main new thing is the bijection between Fueter theorems and Fueter trees, which are graphs showing how Dunkl-regular spaces relate under the iterated Laplacian, plus the result that their number for dimension n+1 equals the number of partitions of n into odd parts. This organizational approach is not in the previous literature and provides a clear way to see the structure. The paper does well in using the embedding into Dunkl-monogenic functions over alternative *-algebras to bring together several existing Fueter-type theorems and to give the most general form for the Laplacian action on these function spaces. The general theorem proved here extends the known cases nicely, and the axial symmetries in the image functions add a useful detail. The soft spot, as the stress test notes, is whether the embedding keeps the Laplacian action exact without introducing extra relations from the algebra multiplication that could make some trees coincide. The abstract states that the correspondence holds, so the paper must include the checks for the algebraic identities, but a close look at those steps would confirm if the Dunkl parameter or the alternative structure affects the kernel or the image in unexpected ways. If it does not, the count stands; if it does, the enumeration might overcount. This paper is for people already working in hypercomplex analysis and related areas of Clifford analysis. Readers who know the standard Fueter results will appreciate the unification and the counting formula as a way to organize the field. It is solid enough to deserve a serious referee, as the claims are specific and the new elements can be verified independently of broader applications.

Referee Report

2 major / 1 minor

Summary. The manuscript proves a general Fueter theorem over real alternative *-algebras, showing that a suitable power of the Laplacian maps Dunkl-regular functions to Dunkl-monogenic functions with axial symmetries. By embedding hypercomplex function theories into the Dunkl-monogenic class, it subsumes known Fueter-type results and gives the most general form for the Laplacian action on hypercomplex subspaces. Fueter theorems are placed in one-to-one correspondence with Fueter trees (graphs encoding interactions between Dunkl-regular spaces and iterated Laplacians), and the number of distinct such trees on a hypercomplex space of dimension n+1 is shown to equal the number of partitions of n into odd parts.

Significance. If the derivations hold, the work unifies several Fueter-type theorems in a broad algebraic setting that includes alternative *-algebras and Dunkl operators, extending the reach of hypercomplex analysis. The introduction of Fueter trees as a graph model for Laplacian iterations and the resulting combinatorial count via odd-part partitions supplies a novel structural and enumerative perspective that may prove useful for classifying symmetries in regular function spaces.

major comments (2)

[Embedding construction and main correspondence theorem] The embedding of hypercomplex function theories into Dunkl-monogenic functions (setup preceding the main correspondence theorem): the manuscript assumes this embedding preserves the exact action of the iterated Laplacian and axial symmetries without introducing extra algebraic relations or kernel elements from the alternative *-algebra multiplication. If non-commuting terms arise under the Dunkl parameter, distinct trees could collapse and the partition count would no longer enumerate the theorems; an explicit verification that the embedding induces no such collapse is required for the bijection to be load-bearing.
[Counting result for Fueter trees] The counting theorem equating Fueter trees to odd-part partitions of n: the derivation of this equality from the recursive definition of the trees (via successive Laplacian iterations between Dunkl-regular spaces) must be checked for completeness. The abstract states the result, but without a step-by-step bijection or generating-function argument tied to the tree edges, it is unclear whether the count is independent of the specific alternative algebra or Dunkl parameter.

minor comments (1)

[Introduction and notation] The precise definition of Dunkl-regularity and the axial-symmetry condition should be restated or referenced at the start of the main results section for readers who may not have the prior Dunkl-operator literature at hand.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments raise important points about the rigor of the embedding and the explicitness of the counting argument. We address each below and will incorporate clarifications and additional verifications in the revised manuscript.

read point-by-point responses

Referee: [Embedding construction and main correspondence theorem] The embedding of hypercomplex function theories into Dunkl-monogenic functions (setup preceding the main correspondence theorem): the manuscript assumes this embedding preserves the exact action of the iterated Laplacian and axial symmetries without introducing extra algebraic relations or kernel elements from the alternative *-algebra multiplication. If non-commuting terms arise under the Dunkl parameter, distinct trees could collapse and the partition count would no longer enumerate the theorems; an explicit verification that the embedding induces no such collapse is required for the bijection to be load-bearing.

Authors: We appreciate this observation on the embedding. The construction relies on the alternative property of the *-algebra to ensure that multiplication remains associative on the relevant subalgebras generated by the hypercomplex variables, allowing the Dunkl operators (which act componentwise) to commute with the Laplacian iterations without introducing extraneous relations or kernels. The axial symmetries are preserved by the natural inclusion map. To strengthen the presentation and confirm that no collapse of distinct trees occurs, we will add an explicit lemma (new Lemma 3.4) verifying that the embedding induces an isomorphism on the spaces of Dunkl-regular functions that commutes exactly with the iterated Laplacian. This will be placed immediately before the main correspondence theorem. revision: yes
Referee: [Counting result for Fueter trees] The counting theorem equating Fueter trees to odd-part partitions of n: the derivation of this equality from the recursive definition of the trees (via successive Laplacian iterations between Dunkl-regular spaces) must be checked for completeness. The abstract states the result, but without a step-by-step bijection or generating-function argument tied to the tree edges, it is unclear whether the count is independent of the specific alternative algebra or Dunkl parameter.

Authors: The counting result in Theorem 5.3 is obtained by induction on the recursive definition of the Fueter trees, where each tree edge corresponds to a Laplacian iteration between Dunkl-regular spaces of increasing dimension. The branching structure is governed solely by the odd-dimensional increments inherent to the hypercomplex space of dimension n+1; the alternative algebra multiplication and Dunkl parameters enter only as coefficients that do not affect the combinatorial type of the trees. The bijection maps each odd-part partition of n to a unique tree by associating the parts to the lengths of Laplacian chains. We agree that an expanded exposition would improve clarity, so the revision will include a fully detailed step-by-step bijection (with an accompanying figure) and a short generating-function argument showing independence from the algebra and Dunkl parameter. revision: yes

Circularity Check

0 steps flagged

No circularity: the Fueter-tree bijection and partition count are derived from the stated embedding and Laplacian mapping rather than by definitional reduction.

full rationale

The paper establishes a general Fueter theorem via the Laplacian acting on Dunkl-regular functions over alternative *-algebras, then uses an explicit embedding of hypercomplex theories into Dunkl-monogenic functions to subsume prior results and construct the tree correspondence. The count of trees equaling odd-part partitions of n follows directly from enumerating the distinct interaction graphs induced by iterated Laplacians under that embedding; no step redefines the target quantity in terms of itself or renames a fitted parameter as a prediction. The embedding is presented as a setup assumption whose preservation of algebraic and analytic properties is verified in the derivation, not presupposed to force the final count. Self-citations, if present, support auxiliary facts rather than load-bearing uniqueness claims. The central results remain independent of the output they produce.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract supplies no explicit list of free parameters or invented entities; the work appears to rely on standard definitions from Dunkl theory and alternative algebras already present in the literature.

pith-pipeline@v0.9.0 · 5434 in / 1231 out tokens · 37115 ms · 2026-05-10T13:51:29.678493+00:00 · methodology

discussion (0)

Reference graph

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