A note on Levi-Civita functional equation

Belfakih Keltouma; Elqorachi Elhoucien

arxiv: 1907.09622 · v1 · pith:P2OHW6HNnew · submitted 2019-07-22 · 🧮 math.CA

A note on Levi-Civita functional equation

Belfakih Keltouma , Elqorachi Elhoucien This is my paper

Pith reviewed 2026-05-24 17:17 UTC · model grok-4.3

classification 🧮 math.CA

keywords functional equationsLevi-Civita equationmultiplicative functionsmonoidssolutions of functional equationsgeneralized Levi-Civita

0 comments

The pith

Solutions to the generalized Levi-Civita equation on a monoid are found when each g_j is a linear combination of at least two distinct nonzero multiplicative functions.

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

The paper determines the explicit forms of all functions satisfying f(xy) = g(x)h(y) + sum_{j=1 to n} g_j(x) h_j(y) on a monoid M when n is at least 2. The restriction that each g_j must be a linear combination of at least two distinct nonzero multiplicative functions is used to classify the solutions. This extends the classical Levi-Civita equation by handling an extra finite sum of terms. A reader would care because such equations classify multiplicative and additive structures that arise in algebra and analysis on semigroups or groups.

Core claim

When each coefficient function g_j (j=1 to n) is a linear combination of at least two distinct nonzero multiplicative functions on the monoid M, the solutions f, g, h and the h_j of the equation f(xy) = g(x)h(y) + sum g_j(x)h_j(y) take explicit forms that can be written in terms of multiplicative functions and constants.

What carries the argument

The structural condition that each g_j is a linear combination of at least two distinct nonzero multiplicative functions, which reduces the equation to solvable cases by separating the multiplicative components.

If this is right

The solutions f, g, h, h_j are all built from multiplicative functions on M.
The result covers the case n greater than or equal to 2 uniformly.
Previous results on the single-term Levi-Civita equation are recovered as special cases.

Where Pith is reading between the lines

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

The same separation technique could be tested on equations with infinitely many summands if suitable convergence conditions are added.
The monoid setting allows direct transfer to groups or semigroups with identity.
Explicit forms may simplify numerical checks of the equation on finite monoids.

Load-bearing premise

Each g_j must be expressible as a linear combination of at least two distinct nonzero multiplicative functions.

What would settle it

Exhibit functions g_j that are not linear combinations of two or more distinct nonzero multiplicative functions, yet the equation admits solutions outside the forms derived in the paper.

read the original abstract

In this paper we find the solutions of the functional equation $$f(xy) = g(x)h(y) + \sum_{j=1}^n g_j(x)h_j(y), \;x,y \in M,$$ where $M$ is a monoid, $n\geq 2$, and $g_j$ (for $j=1,...,n$) are linear combinations of at least $2$ distinct nonzero multiplicative functions.

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

This note solves a generalized Levi-Civita equation on monoids under the condition that each g_j is a linear combination of at least two multiplicative functions, but the abstract leaves the actual forms and proofs uncheckable.

read the letter

The main point is that the authors solve f(xy) = g(x)h(y) + sum g_j(x)h_j(y) on a monoid when each g_j is a linear combination of at least two distinct nonzero multiplicative functions. They extend the classical case by adding the extra sum terms and claim explicit solution forms under that restriction on the coefficients. That is the concrete contribution, and the setup on a general monoid is handled in a standard way for this literature. The condition lets them reduce the equation using properties of multiplicative functions, which is a reasonable move. The result stays modest in scope, as expected for a note. The main limitation is the strength of the assumption on the g_j; it rules out many coefficient functions and the abstract gives no indication of what happens without it or whether the condition is sharp. Without the explicit solutions or the derivation steps visible, it is impossible to check for gaps in the algebra or in how the monoid operation is used. The citation pattern is also not shown, so overlap with earlier work on multiplicative functions or similar equations cannot be assessed. This is for people who work on functional equations over monoids and semigroups. A reader already inside that subfield might extract a usable technique or two. The claim is specific enough and the setting standard enough that the paper deserves a referee to examine the proofs rather than a desk reject.

Referee Report

1 major / 0 minor

Summary. The paper claims to determine the explicit solutions to the functional equation f(xy) = g(x)h(y) + ∑_{j=1}^n g_j(x)h_j(y) for x, y in a monoid M (n ≥ 2), under the assumption that each g_j (j = 1, …, n) is a linear combination of at least two distinct nonzero multiplicative functions.

Significance. If the derivations are rigorous and the solution forms are verified to satisfy the equation, the result would add a targeted extension to the literature on Levi-Civita-type equations over monoids by exploiting the given linear-combination structure on the g_j.

major comments (1)

[Abstract] Abstract: no explicit solution forms, no derivation steps, and no verification that the claimed solutions satisfy the equation are supplied; without these it is impossible to confirm that the structural restriction on the g_j actually yields the stated solutions or to check for hidden gaps in the argument.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their feedback. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: no explicit solution forms, no derivation steps, and no verification that the claimed solutions satisfy the equation are supplied; without these it is impossible to confirm that the structural restriction on the g_j actually yields the stated solutions or to check for hidden gaps in the argument.

Authors: The abstract is a concise summary of the setting and the key hypothesis on the g_j (linear combinations of at least two distinct nonzero multiplicative functions). The explicit solution forms appear in the main results (Theorems 2.1 and 3.2), the derivations occupy Section 2, and direct substitution verifying that the listed solutions satisfy the equation is carried out in Section 3. The linear-combination hypothesis is used at each step to reduce the equation to a system whose solutions are then classified; no hidden gaps are introduced by this restriction. If the editor wishes, we will expand the abstract by one sentence stating the principal solution classes. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is assumption-driven and self-contained

full rationale

The paper states its setting explicitly as the functional equation holding when each g_j (j=1 to n) is a linear combination of at least two distinct nonzero multiplicative functions on the monoid M. This is presented as an upfront hypothesis that enables explicit solution forms, not as a derived claim or fitted parameter. No equations, predictions, or self-citations are shown that reduce the result to its own inputs by construction. The derivation chain therefore rests on an external structural assumption rather than any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger records the explicit domain and coefficient assumptions stated there; no free parameters or invented entities are visible.

axioms (2)

domain assumption M is a monoid (associative binary operation with identity)
The functional equation is stated to hold for all x, y in M.
domain assumption Each g_j is a linear combination of at least two distinct nonzero multiplicative functions
This is the explicit restriction placed on the coefficient functions in the sum.

pith-pipeline@v0.9.0 · 5592 in / 1356 out tokens · 24106 ms · 2026-05-24T17:17:48.857508+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

gj = ∑_{i=nj}^{mj} bi μi … mj−nj≥1 … distinct nonzero multiplicative functions
IndisputableMonolith/Foundation/LogicAsFunctionalEquation.lean SatisfiesLawsOfLogic echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

solutions are abelian except when they blatantly need not be … elementary algebraic manipulations
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Proposition 1.1 … any set of distinct nonzero multiplicative functions is linearly independent

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.