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arxiv: 2501.01386 · v1 · submitted 2025-01-02 · 🧮 math.CV

Solutions of systems of certain Fermat-type PDDEs

Raju Biswas , Rajib Mandal This is my paper

Pith reviewed 2026-05-23 06:32 UTC · model grok-4.3

classification 🧮 math.CV
keywords Fermat-type equationspartial differential-difference equationsentire solutionsmeromorphic solutionsfinite orderseveral complex variablesvalue distribution theory
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The pith

Pairs of finite-order entire and meromorphic solutions exist for certain Fermat-type PDDE systems 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 examines systems of Fermat-type partial differential-difference equations involving several complex variables. It proves existence of pairs of finite-order entire and meromorphic solutions and supplies their explicit forms. These results refine and extend earlier findings on similar functional equations by applying growth estimates to limit solution possibilities. A reader would care because the explicit forms clarify which algebraic combinations of functions can satisfy the equations under order constraints.

Core claim

For the systems under consideration, pairs of finite-order entire functions and meromorphic functions exist that satisfy the Fermat-type relations, and the paper determines the explicit expressions these solution pairs must take.

What carries the argument

Growth estimates from Nevanlinna theory in several complex variables, used to constrain the possible forms of finite-order solutions.

Load-bearing premise

Solutions are restricted to finite order so that growth estimates can be applied to force them into specific algebraic shapes.

What would settle it

Discovery of a finite-order entire or meromorphic solution pair to one of the studied systems that does not match any of the explicit forms listed in the paper.

read the original abstract

The objective of this paper is to investigate the existence and the forms of the pair of finite order entire and meromorphic solutions of some certain systems of Fermat-type partial differential-difference equations of several complex variables. These results represent some refinements and generalizations of the earlier findings, especially the results due to Xu {\it et al.} (J. Math. Anal. Appl. 483(2) (2020)). We provide some examples to support the 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.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates the existence and explicit forms of pairs of finite-order entire and meromorphic solutions to systems of Fermat-type partial differential-difference equations (PDDEs) in several complex variables. It presents refinements and generalizations of results due to Xu et al. (J. Math. Anal. Appl. 483(2) (2020)), derives candidate solution forms via Nevanlinna-theoretic growth estimates under the finite-order hypothesis, and supplies analytic verifications together with supporting examples.

Significance. If the central derivations hold, the work contributes concrete explicit solution pairs and growth-order constraints for Fermat-type PDDE systems, extending single-variable results to several variables. The provision of both analytic proofs and concrete examples is a strength that supports reproducibility and further study in the area.

minor comments (3)
  1. [Abstract] The abstract refers to 'certain systems' without naming the precise equations; a brief explicit statement of the systems studied in the main theorems would improve readability.
  2. [Main results] In the growth-estimate arguments, the transition from the finite-order hypothesis to the explicit forms would benefit from an additional sentence clarifying how the several-variable Nevanlinna lemmas are applied (e.g., which lemma controls the proximity function for the difference operators).
  3. [Examples] The examples section would be strengthened by a short table or paragraph recording the computed orders of the exhibited solutions to confirm they satisfy the finite-order premise.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. The assessment accurately reflects the manuscript's focus on finite-order solutions to Fermat-type PDDE systems and its relation to Xu et al. (2020). Since no specific major comments appear in the report, we have no point-by-point items to address.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard Nevanlinna estimates

full rationale

The paper states the Fermat-type PDDE systems explicitly, assumes finite order upfront, and applies standard growth lemmas from Nevanlinna theory to derive candidate solution forms that are then verified analytically and by examples. No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation; the cited prior work (Xu et al.) is external and the proofs remain independent of the target conclusions. This is the normal case of a self-contained analytic derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard results from Nevanlinna theory for meromorphic functions of finite order and basic properties of entire functions; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard Nevanlinna theory growth estimates apply to finite-order meromorphic functions in several complex variables
    Invoked implicitly when classifying solutions by order and type

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discussion (0)

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Reference graph

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