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arxiv: 2504.01359 · v4 · pith:AII4XGREnew · submitted 2025-04-02 · 🧮 math.CV

Monogenic functions over real alternative *-algebras: fundamental results and applications

Qinghai Huo , Guangbin Ren , Zhenghua Xu This is my paper

Pith reviewed 2026-05-22 22:33 UTC · model grok-4.3

classification 🧮 math.CV
keywords monogenic functionsalternative algebrasCauchy-Pompeiu formulaTaylor serieshypercomplex analysisnon-associativityquaternionsoctonions
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The pith

Monogenic functions over real alternative star-algebras satisfy Cauchy-Pompeiu integral formulas and Taylor expansions despite non-commutativity and non-associativity.

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

The paper defines monogenic functions over real alternative star-algebras to unify quaternionic, octonionic, and Clifford analysis. It derives a Cauchy-Pompeiu integral formula and Taylor series expansions that hold in hypercomplex subspaces even though multiplication is neither commutative nor associative. A sympathetic reader would care because the same integral and series tools that work in complex analysis now apply across these broader algebraic structures without separate case-by-case proofs.

Core claim

Monogenic functions over real alternative *-algebras admit a Cauchy-Pompeiu integral formula and Taylor series expansion in hypercomplex subspaces, with full consideration of non-commutativity and non-associativity of the multiplication.

What carries the argument

The monogenicity condition on functions valued in real alternative *-algebras, which is used to establish integral representations and power series despite lack of associativity.

If this is right

  • The integral formula and series expansions apply uniformly to quaternionic, octonionic, and Clifford settings as special cases.
  • Hypercomplex analysis results that rely on these formulas carry over directly to the general alternative algebra setting.
  • Functions satisfying the monogenicity condition possess local power series representations in suitable subspaces.

Where Pith is reading between the lines

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

  • The same approach might allow integral formulas in other non-associative algebras if an analogous monogenicity condition can be identified.
  • Numerical methods based on the Cauchy-Pompeiu formula could be extended to computations over octonions and similar algebras.

Load-bearing premise

The definition of monogenicity is strong enough to imply the classical integral and series properties when multiplication is neither commutative nor associative.

What would settle it

An explicit monogenic function in the octonions for which the Cauchy-Pompeiu formula fails to recover the function values.

read the original abstract

The concept of monogenic functions over real alternative $\ast$-algebras has recently been introduced to unify several classical monogenic (or regular) functions theories in hypercomplex analysis, including quaternionic, octonionic, and Clifford analysis. This paper explores the fundamental properties of these monogenic functions, focusing on the Cauchy-Pompeiu integral formula and Taylor series expansion in hypercomplex subspaces, among which the non-commutativity and especially non-associativity of multiplications demand full considerations. The theory presented herein provides a robust framework for understanding monogenic functions in the context of real alternative $\ast$-algebras, shedding light on the interplay between algebraic structures and hypercomplex analysis.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces monogenic functions over real alternative *-algebras as a unifying framework for quaternionic, octonionic, and Clifford analysis. It claims to establish the Cauchy-Pompeiu integral formula and Taylor series expansions in hypercomplex subspaces while fully accounting for non-commutativity and non-associativity of the multiplication.

Significance. If the derivations hold, the work would offer a significant generalization of hypercomplex analysis to non-associative settings such as octonions. The explicit treatment of non-associativity is a potential strength, but the central claims rest on whether the monogenicity condition suffices for the integral and series representations without associativity.

major comments (2)
  1. [Cauchy-Pompeiu formula section] The derivation of the Cauchy-Pompeiu formula must explicitly justify all steps that classically rely on associativity (e.g., product rules for differentials or kernel evaluations on the boundary) using only alternativity of the algebra; without such justification the extension to non-associative cases is not automatic.
  2. [Definition of monogenicity and main theorems] The definition of monogenicity (presumably via a Dirac-type operator D) needs to be shown to imply the integral representation and power series even when the associator is nonzero; the manuscript should supply a concrete verification or explicit use of alternativity identities for the octonion case.
minor comments (1)
  1. [Abstract] The abstract states the existence of the formulas but supplies no derivation outline or error estimates; adding a brief indication of the key technical steps would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the treatment of alternativity requires greater explicitness. We address each major comment below and will revise the manuscript to strengthen the justifications.

read point-by-point responses

  1. Referee: [Cauchy-Pompeiu formula section] The derivation of the Cauchy-Pompeiu formula must explicitly justify all steps that classically rely on associativity (e.g., product rules for differentials or kernel evaluations on the boundary) using only alternativity of the algebra; without such justification the extension to non-associative cases is not automatic.

    Authors: We agree that the steps must be justified solely from alternativity. The manuscript already invokes alternativity identities (such as the vanishing of the associator in specific configurations and the alternative law) when deriving the product rule for the Dirac operator and when evaluating the kernel on the boundary. However, these invocations are currently implicit. In the revision we will add an auxiliary lemma that isolates each classically associativity-dependent step and replaces it with the corresponding alternativity identity, thereby making the argument self-contained for non-associative algebras. revision: yes

  2. Referee: [Definition of monogenicity and main theorems] The definition of monogenicity (presumably via a Dirac-type operator D) needs to be shown to imply the integral representation and power series even when the associator is nonzero; the manuscript should supply a concrete verification or explicit use of alternativity identities for the octonion case.

    Authors: The definition of monogenicity is formulated via the Dirac operator D on the alternative *-algebra, and the proofs of the integral formula and Taylor expansion rely on alternativity to ensure that associator terms cancel. To meet the referee’s request for concrete verification, the revised manuscript will include a short subsection that specializes the general argument to the octonions, explicitly tracking the nonzero associator and confirming that it does not affect the final identities because of the alternativity relations. revision: yes

Circularity Check

0 steps flagged

No circularity: standard definitional extension with independent proofs

full rationale

The provided abstract and context describe a definition of monogenicity over alternative *-algebras followed by derivation of Cauchy-Pompeiu and Taylor results. No quoted equations or self-citations reduce any claimed result to a fitted input or prior self-result by construction. The derivation chain is presented as extending classical identities while accounting for non-associativity, without self-definitional loops or load-bearing self-citations. This is the expected non-circular outcome for a foundational math paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are identifiable.

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discussion (0)

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    math.CV 2026-04 unverdicted novelty 6.0

    Fueter theorems over alternative star-algebras correspond one-to-one with Fueter trees whose number on an (n+1)-dimensional hypercomplex space equals the number of partitions of n into odd parts.

  2. Monogenic functions over real alternative *-algebras: the several hypercomplex variables case

    math.CV 2025-06 unverdicted novelty 6.0

    Introduces monogenic functions of several hypercomplex variables over real alternative *-algebras and establishes Bochner-Martinelli formula, Plemelj-Sokhotski formula, and Hartogs extension theorem.

  3. Monogenic functions over real alternative *-algebras: the several hypercomplex variables case

    math.CV 2025-06 unverdicted novelty 6.0

    Initiates monogenic functions of several hypercomplex variables over real alternative *-algebras and establishes Bochner-Martinelli, Plemelj-Sokhotski, and Hartogs extension results in this unified setting.

Reference graph

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