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arxiv: 2503.12409 · v4 · pith:4VRLCID7new · submitted 2025-03-16 · 🧮 math.CV

Generalized partial-slice monogenic functions: the octonionic case

Zhenghua Xu , Irene Sabadini This is my paper

Pith reviewed 2026-05-23 00:41 UTC · model grok-4.3

classification 🧮 math.CV
keywords generalized partial-slice monogenic functionsoctonionsslice regular functionsregular functionsidentity theoremCauchy integral formulamaximum modulus principleFueter polynomials
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The pith

Generalized partial-slice monogenic functions over octonions include both regular and slice regular functions as special cases.

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

This paper introduces generalized partial-slice monogenic functions in the octonionic setting. The new class contains the earlier notions of regular functions and slice regular functions over octonions. The authors establish an identity theorem, a representation formula, Cauchy-type integral formulas, the maximum modulus principle, Fueter polynomials, and Taylor series expansions for these functions. These results extend the theory of monogenic functions from associative Clifford algebras to the non-associative octonions. A sympathetic reader would care because the approach provides a unified framework for analytic properties in alternative algebras.

Core claim

The generalized partial-slice monogenic functions over octonions encompass the regular functions and the slice regular functions. In this non-associative setting the standard properties of monogenic functions continue to hold, including the identity theorem, representation formula, Cauchy and Cauchy-Pompeiu integral formulas, maximum modulus principle, Fueter polynomials and Taylor series expansions. The results are shown to apply more generally to real alternative algebras with a conjugation.

What carries the argument

The generalized partial-slice monogenic function, defined by extending the partial-slice condition from Clifford algebras to octonions.

If this is right

  • Both regular and slice regular functions satisfy the new definition.
  • The identity theorem holds for these functions.
  • Cauchy and Cauchy-Pompeiu integral formulas represent the functions via boundary integrals.
  • The maximum modulus principle applies.
  • Taylor series expansions and Fueter polynomials are available.

Where Pith is reading between the lines

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

  • The same construction and proofs should apply to other real alternative algebras equipped with conjugation.
  • This may allow results from the Clifford setting to transfer directly to octonion-valued problems in geometry or physics.
  • Further specialization of the partial-slice parameter could recover additional subclasses of functions.

Load-bearing premise

The algebraic properties of octonions as an alternative algebra with conjugation suffice for the proofs to go through without modification from the Clifford algebra case.

What would settle it

A function that meets the generalized partial-slice monogenic condition over octonions but violates the identity theorem, such as a non-zero function whose zero set has an accumulation point.

read the original abstract

In a recent paper [Trans. Amer. Math. Soc. 378 (2025), 851-883], the concept of generalized partial-slice monogenic (or regular) function was introduced over Clifford algebras. The present paper shall extend the study of generalized partial-slice monogenic functions from the associative case of Clifford algebras to non-associative alternative algebras, such as octonions. The new class of functions encompasses the regular functions [Rend. Sem. Mat. Univ. Padova 50 (1973), 251-267] and slice regular functions [Rocky Mountain J. Math. 40 (2010), no. 1, 225-241] over octonions, indeed both appear in the theory as special cases. In the non-associative setting of octonions, we shall develop some fundamental properties such as identity theorem, Representation Formula, Cauchy (and Cauchy-Pompeiu) integral formula, maximum modulus principle, Fueter polynomials, Taylor series expansion. As a complement, the paper also introduces and discusses the notion of generalized partial-slice (and regular) functions. Although the study is limited to the case of octonions, it is clear from the statements and the arguments in the proofs that the results hold more in general in real alternative algebras equipped with a notion of conjugation.

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 / 2 minor

Summary. The paper extends the notion of generalized partial-slice monogenic functions, previously defined over Clifford algebras, to the octonions. It defines a new class that includes both regular functions and slice-regular functions over octonions as special cases, and establishes several fundamental results in this non-associative setting: the identity theorem, Representation Formula, Cauchy and Cauchy-Pompeiu integral formulas, maximum modulus principle, Fueter polynomials, and Taylor series expansions. The authors state that the same arguments apply verbatim to any real alternative algebra equipped with a conjugation.

Significance. If the proofs are shown to rely only on alternativity rather than full associativity, the work would supply a coherent framework that unifies known classes of octonionic monogenic functions and extends naturally to other alternative algebras. The explicit recovery of both regular and slice-regular functions as special cases is a concrete strength.

major comments (2)
  1. [Abstract] Abstract and final paragraph: the claim that 'the statements and the arguments in the proofs' carry over unchanged to arbitrary real alternative algebras with conjugation is load-bearing for the paper's scope. No explicit check is supplied that steps involving products inside integrals, power-series multiplication, or Fueter-polynomial recursions use only alternativity (xy)x = x(yx) and never require (xy)z = x(yz).
  2. [Definition section] Definition of generalized partial-slice monogenic function (presumably §2): the partial-slice condition must be formulated so that the non-associativity of octonion multiplication does not invalidate the subsequent integral representations or the Representation Formula; the manuscript does not isolate the precise algebraic identity used at each step.
minor comments (2)
  1. [Preliminaries] Notation for the octonion conjugation and the splitting into real and imaginary parts should be stated once at the beginning and used consistently.
  2. [References] The 2025 Trans. Amer. Math. Soc. reference should include the full bibliographic details (volume, pages, DOI) in the reference list.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments, which point to useful clarifications on the algebraic foundations of the results.

read point-by-point responses

  1. Referee: [Abstract] Abstract and final paragraph: the claim that 'the statements and the arguments in the proofs' carry over unchanged to arbitrary real alternative algebras with conjugation is load-bearing for the paper's scope. No explicit check is supplied that steps involving products inside integrals, power-series multiplication, or Fueter-polynomial recursions use only alternativity (xy)x = x(yx) and never require (xy)z = x(yz).

    Authors: We agree that an explicit verification would strengthen the manuscript. In the revised version we will insert a dedicated remark (or short subsection) that systematically identifies the algebraic identities used in each proof—identity theorem, Representation Formula, integral formulas, maximum modulus principle, Fueter polynomials, and Taylor series—and confirms that only the alternative laws are invoked, without any appeal to full associativity. revision: yes

  2. Referee: [Definition section] Definition of generalized partial-slice monogenic function (presumably §2): the partial-slice condition must be formulated so that the non-associativity of octonion multiplication does not invalidate the subsequent integral representations or the Representation Formula; the manuscript does not isolate the precise algebraic identity used at each step.

    Authors: The partial-slice condition is stated so that the subsequent formulas remain valid under alternativity. To make this transparent we will revise the definition section (and the proofs that follow) to explicitly name the precise identities—primarily the left and right alternative laws—employed at each step when deriving the integral representations and the Representation Formula. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; new definition and proofs for octonions are independent of the cited Clifford-algebra source

full rationale

The paper cites a 2025 Trans. AMS paper solely to introduce the generalized partial-slice monogenic concept in the associative Clifford setting, then defines an analogous class for octonions and derives its properties (identity theorem, Representation Formula, Cauchy integral formula, Fueter polynomials, Taylor expansion) directly in the non-associative case. The abstract explicitly states that both regular and slice-regular functions appear as special cases within the new theory, and the proofs are presented as self-contained for octonions. The further claim that the same arguments apply verbatim to any real alternative algebra with conjugation is an assertion about the scope of the given proofs rather than a reduction of any equation or result to the prior paper by construction. No self-definitional loop, fitted-input prediction, or load-bearing self-citation chain is exhibited; the derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities beyond the new function class itself; the central claim rests on standard properties of octonions and alternative algebras that are assumed known from prior literature.

invented entities (1)
  • generalized partial-slice monogenic function over octonions no independent evidence
    purpose: To unify regular and slice regular functions and support the listed analytic properties in the non-associative setting
    This is the central new object introduced by the paper; no independent falsifiable evidence is supplied in the abstract.

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

    math.CV 2025-04 unverdicted novelty 7.0

    Presents integral formulas and series expansions for monogenic functions in real alternative *-algebras that unify several hypercomplex analysis theories.

  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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