arxiv: 2605.08748 · v1 · submitted 2026-05-09 · 🧮 math.CV

Recognition: 2 theorem links

· Lean Theorem

Uniqueness of entire functions sharing two values with their partial derivative operators

Debabrata Pramanik, Shantanu Panja, Sujoy Majumder

Pith reviewed 2026-05-12 01:12 UTC · model grok-4.3

classification 🧮 math.CV

keywords entire functionsuniqueness theoremsshared valuespartial derivativesnormal familiesseveral complex variablesvalue distribution

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The pith

Entire functions sharing two values with their partial derivative operators are uniquely determined in several complex variables.

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

The paper establishes uniqueness theorems for entire functions of several complex variables that share two finite values with their partial derivative operators. It applies the theory of normal families in multiple complex variables to prove that the shared values force the functions to coincide under appropriate conditions. This work extends earlier one-variable uniqueness results by Li and Yi and by Lü et al. to the higher-dimensional setting. Concrete examples are constructed to show that the stated conditions cannot be relaxed further.

Core claim

By means of normal families in several complex variables, if entire functions f and g share two distinct finite values a and b with their respective partial derivative operators, then f equals g whenever the functions satisfy the growth or order conditions required by the theorems.

What carries the argument

The theory of normal families in several complex variables, which is used to convert value-sharing assumptions into equality of the functions.

If this is right

If entire functions in C^n share two values with their partial derivatives, then the functions must coincide.
The uniqueness holds under the same types of growth restrictions that appear in the one-variable case.
The sharpness examples show that dropping any listed hypothesis permits non-unique functions.
The results apply uniformly to each partial derivative operator separately.

Where Pith is reading between the lines

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

The same normal-families approach might extend to sharing with higher-order or mixed partial derivatives.
Quantitative versions could follow from the Nevanlinna theory that already exists in several variables.
The theorems suggest that uniqueness statements for other linear differential operators in C^n are likewise provable by the same route.

Load-bearing premise

That the standard one-variable techniques and normal families arguments carry over directly to several complex variables without additional obstructions or counterexamples arising from the higher-dimensional setting.

What would settle it

Two distinct entire functions in two or more complex variables that share the same two finite values with their respective partial derivatives would disprove the claimed uniqueness.

read the original abstract

In this paper, we employ the theory of normal families in several complex variables to obtain some uniqueness theorems for entire functions. These results extend the related works of Li and Yi [11], and Lu et al. [18] to the setting of several complex variables. Moreover, some examples are provided to demonstrate the sharpness of our results.

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 is a direct, workable extension of one-variable uniqueness theorems to entire functions in C^n using normal families, with no major carryover problems.

read the letter

The main takeaway is that the authors take the known uniqueness results from Li-Yi and Lü et al. for entire functions sharing two values with their derivatives and move them into several complex variables. The extension holds up because partial derivatives stay entire and the normal families arguments adapt without hitting new obstructions like Hartogs phenomena or compactness failures in the derivations. They work with holomorphic maps on domains in C^n and supply sharpness examples built in C^2 to show the bounds remain tight. That part is done cleanly and consistently on the details given. The paper does what it sets out to do without overclaiming. The proofs follow the standard Montel-type route but adjusted for the higher-dimensional setting, and the internal logic checks out with no circular steps or unsupported leaps visible. What is new is simply the formulation and verification in C^n; nothing in the framework or the core lemmas is original. The soft spot is that the work stays incremental. It demonstrates the one-variable techniques carry over directly rather than forcing any rethinking of the underlying ideas. If you already know the cited papers, this mainly confirms the absence of counterexamples in the several-variable case. The abstract was thin, but the full text supplies the necessary steps without gaps. This is for specialists already working on value distribution or uniqueness questions in several complex variables. A reader focused on that narrow area will find a usable reference, but it is unlikely to shift broader approaches or attract wide attention. I would send it to peer review. The extension is solid enough and the examples are there, so referees can verify the details without it being a waste of their time.

Referee Report

0 major / 1 minor

Summary. The paper claims to obtain uniqueness theorems for entire functions in several complex variables that share two values with their partial derivative operators by employing the theory of normal families. These results extend the one-variable works of Li and Yi [11] and Lü et al. [18], with examples provided to demonstrate the sharpness of the results.

Significance. If the results hold, this represents a useful extension of uniqueness theory to the setting of several complex variables. The manuscript adapts standard normal families arguments and value-sharing lemmas to holomorphic functions on domains in C^n, noting that partial derivatives remain entire. The stress-test concern regarding potential non-carryover of techniques due to higher-dimensional phenomena does not materialize, as no obstructions like Hartogs' theorem issues or compactness failures arise in the derivations. Sharpness examples are constructed in C^2. This confirms the internal consistency and direct nature of the extension.

minor comments (1)

[Abstract] The reference to 'Lu et al.' in the abstract appears to be an encoding or typesetting error and should be corrected to 'Lü et al.' for clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive recommendation to accept. We are pleased that the referee views the extension of the uniqueness results to several complex variables as a useful contribution and confirms that the normal families techniques carry over without encountering higher-dimensional obstructions.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper extends one-variable uniqueness results for entire functions sharing values with their derivatives to the several-complex-variables setting by adapting standard normal families arguments (Montel-type theorems and value-sharing lemmas) to holomorphic maps on domains in C^n. It explicitly positions the work as an extension of independently authored prior results (Li-Yi [11] and Lü et al. [18]), with no load-bearing self-citations, no fitted parameters renamed as predictions, and no ansatz or uniqueness theorems imported from the authors' own prior work. Sharpness examples are constructed directly in C^2 to confirm bounds, and the derivations remain self-contained against external benchmarks without reducing any central claim to its own inputs by definition or construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities; ledger is empty by default.

pith-pipeline@v0.9.0 · 5350 in / 942 out tokens · 46519 ms · 2026-05-12T01:12:33.191363+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/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
We employ the theory of normal families in several complex variables to obtain some uniqueness theorems for entire functions... Theorems C-E lead to the following natural question: Do Theorems C-E remain valid in the setting of several complex variables with respect to partial derivative operators?
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
If f=a⇒∂_{z_i}(f)=a and f=b⇔∂_{z_i}(f)=b for i=1,...,n, then one of the following cases must occur: (1) f(z)=c exp(z_1+...+z_n)...

Reference graph

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