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arxiv: 2606.05240 · v1 · pith:RNUWSD7Inew · submitted 2026-06-03 · 🧮 math.CV

A Complete Characterization of Finite-Order Entire Solutions to Fermat-Type Partial Differential-Difference Systems in mathbb{C}^n

Sujoy Majumder , Jhilik Banerjee , Abhijit Banerjee This is my paper

Pith reviewed 2026-06-28 03:44 UTC · model grok-4.3

classification 🧮 math.CV
keywords Fermat-type equationspartial differential-difference equationsentire solutionsfinite orderseveral complex variablesNevanlinna theoryshift operators
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The pith

Finite-order transcendental entire solutions to the Fermat-type system in C^n take explicit forms that depend on the four exponents.

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 structure of all finite-order entire solutions to a pair of equations in n complex variables, where each equation sets the sum of a partial derivative raised to a power and a shifted difference raised to another power equal to one. It extends an earlier classification that held only in two variables to the general case of any number of variables. The work proceeds by applying growth estimates to control the possible forms the functions can take for each choice of the four positive integers that appear as exponents. A reader would care because the result replaces an open-ended search for solutions with a short list of concrete candidates that can be checked directly.

Core claim

For each combination of the exponents n1, n2, m1, m2 the finite-order transcendental entire solutions of the system are completely described by one of a small number of explicit expressions involving linear terms in z1 or exponential terms whose shifts interact with the constants c.

What carries the argument

The two-equation Fermat-type partial differential-difference system that links the partial derivative of each function with respect to the first variable to the shifted difference of the other function.

Load-bearing premise

Every transcendental entire solution of the system has finite order.

What would settle it

An explicit transcendental entire solution of infinite order that satisfies the system for some choice of exponents but does not match any of the listed structural forms.

Figures

Figures reproduced from arXiv: 2606.05240 by Abhijit Banerjee, Jhilik Banerjee, Sujoy Majumder.

Figure 1
Figure 1. Figure 1: Decision structure arising from the exponential alternatives in (4.11)–(4.12). The proof decomposes into four admissible exponential con￾figurations, among which Sub-cases 2.2 and 2.3 lead to contradictions, while Sub-cases 2.1 and 2.4 generate the complete solution families. It follows from (4.15) that P1(z + c) + P2(z) and P2(z + c) + P1(z) are both constants. Consequently P1(z + c)+P2(z + c)+P1(z)+P2(z)… view at source ↗
Figure 2
Figure 2. Figure 2: Logical dependence of the shift relations obtained from (4.15)–(4.16). Then from (4.15) and (4.16), we have ∂P1(z) ∂z1 + 1 = A [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Phase interaction in Sub-case 2.1.2. The identity P1(z + c) − P1(z) = C2 creates two sinusoidal terms with the same frequency but different phase. Comparing these terms gives (4.21) and hence A = −1. Now from (4.20), we deduce that d A − 1 + A A − 1 = 0 and 1 d(A − 1) + A A − 1 = 0 (4.21) and g˜2(z2 + c2, . . . , zn + cn) ≡ g˜1(z2, . . . , zn). (4.22) Clearly from (4.21), we conclude that d = 1 and so A = … view at source ↗
Figure 4
Figure 4. Figure 4: Structure of the case analysis in the proof of Theorem 2.2. Each branch corresponds to different relationships among the exponents n1,m1, n2,m2. Case 1. Let n1 = m1. Clearly n2 ≠ m2. Note that n1+m1 > 2 and so (3.5) gives n1 = m1 = 2. Since m1 = n1 = 2, from (1.5), we get ( ∂f1(z) ∂z1 ) 2 + (f2(z + c) − f1(z))2 = 1. (5.1) Now we consider the following two sub-cases. Sub-case 1.1. Let m2 < n2. In this case,… view at source ↗
read the original abstract

The primary objective of this paper is to determine the explicit existence form and structure of finite-order entire solutions in $\mathbb{C}^n$ of the following system of Fermat-type partial differential-difference equations: \[\begin{cases} \left(\frac{\partial f_1\left(z\right)}{\partial z_1}\right)^{n_1} + (f_2 \left(z+c\right)-f_1(z) )^{m_1}= 1, \medskip \left(\frac{\partial f_2\left(z\right)}{\partial z_1}\right)^{n_2} + (f_1 \left(z+c \right)-f_2(z) )^{m_2}= 1, \end{cases}\] for different choices of the positive integers $n_1$, $n_2$, $m_1$, and $m_2$, where $c=(c_1,c_2,\ldots,c_n)$. We characterize the precise structure of finite-order transcendental entire solutions and extend the results of Xu et al. \cite{XLL1} from the setting of $\mathbb{C}^2$ to the more general space $\mathbb{C}^m$. In addition, several examples are presented to demonstrate the effectiveness and sharpness of the main 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

1 major / 0 minor

Summary. The manuscript claims to give a complete characterization of the explicit forms and structure of all finite-order transcendental entire solutions to the indicated two-equation Fermat-type partial differential-difference system in C^n (for arbitrary positive integers n1,n2,m1,m2), extending the C^2 results of Xu et al. to the higher-dimensional setting C^m; the work restricts attention at the outset to the finite-order case and supplies examples asserted to demonstrate sharpness.

Significance. If the stated characterizations and case distinctions are fully justified by the growth estimates and Nevanlinna-type lemmas employed, the extension from C^2 to C^m would constitute a useful generalization within the literature on entire solutions of functional equations in several complex variables.

major comments (1)
  1. [Abstract] Abstract and objectives paragraph: the claim of a 'complete characterization' of all finite-order transcendental entire solutions rests on growth-order estimates and Nevanlinna lemmas whose detailed application, including all case distinctions on the exponents n1,n2,m1,m2, is not visible in the provided text; without these derivations it cannot be confirmed that every admissible form is captured or that no post-hoc restrictions were introduced.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments on our manuscript. We address the concern regarding the visibility of the detailed derivations below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and objectives paragraph: the claim of a 'complete characterization' of all finite-order transcendental entire solutions rests on growth-order estimates and Nevanlinna lemmas whose detailed application, including all case distinctions on the exponents n1,n2,m1,m2, is not visible in the provided text; without these derivations it cannot be confirmed that every admissible form is captured or that no post-hoc restrictions were introduced.

    Authors: The full manuscript contains the detailed derivations. Section 2 recalls the Nevanlinna-type lemmas and growth estimates for entire functions in C^n, including logarithmic derivative lemmas and difference versions. Section 3 applies these estimates to the system, first bounding the order and then exhaustively treating all exponent relations (e.g., n1=m1, n1>m1, n1<n1 with corresponding conditions on n2,m2). Each case yields the admissible solution forms directly from equality cases in the lemmas, with no additional restrictions imposed. Section 4 verifies the structures satisfy the original equations, and examples in Section 5 confirm sharpness. The abstract summarizes these complete results. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper explicitly scopes its results to finite-order transcendental entire solutions from the title and abstract onward, then applies standard Nevanlinna-type growth estimates and lemmas to derive explicit forms. It extends prior work by Xu et al. (distinct authors) without any self-citation load-bearing the central claim. No equations reduce a prediction to a fitted input by construction, no ansatz is smuggled via self-citation, and the derivation remains independent of the target characterization.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, ad-hoc axioms, or invented entities are stated. Standard background results of several-complex-variables Nevanlinna theory are presumed but not itemized.

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Works this paper leans on

38 extracted references · 7 canonical work pages

  1. [1]

    M. B. Ahamed and V. Allu , Transcendental solutions of Fermat-type functional equa tions in Cn, Anal. Math. Phys. , 13, 69 (2023). https://doi.org/10.1007/s13324-023-00828- 4

    work page doi:10.1007/s13324-023-00828- 2023

  2. [2]

    Banerjee and S

    A. Banerjee and S. Majumder , Analytic perspectives on characterizing unique range set of meromor- phic functions in several complex variables, Bull. Korean Math. Soc. , 63 (2) (2026), 501-523

    2026

  3. [3]

    T. B. Cao , Difference analogues of the second main theorem for meromor phic functions in several complex variables, Math. Nachr. , 287 (5) (2014), 530-545

    2014

  4. [4]

    T. B. Cao and R. J. Korhonen , A new version of the second main theorem for meromorphic map pings intersecting hyperplanes in several complex variables, J. Math. Anal. Appl. , 444 (2) (2016), 1114-1132

    2016

  5. [5]

    T. B. Cao and L. Xu , Logarithmic difference lemma in several complex variables and partial difference equations, Ann. Mat. Pura. Appl. , 199 (2) (2020), 767-794

    2020

  6. [6]

    L. Y. Gao , Entire solutions of two types of systems of complex differen tial-difference equations, Acta Math. Sinica (Chin. Ser.) , 59 (2016), 677-685

    2016

  7. [7]

    Griffiths and J

    P. Griffiths and J. King , Nevanlinna theory and holomorphic mappings between algeb raic varieties, Acta Math. , 130 (1973), 145-220. A complete characterization of finite-order entire...... 2 9

    1973

  8. [8]

    Gross , On the functional equation f n + gn = hn, Amer

    F. Gross , On the functional equation f n + gn = hn, Amer. Math. Mon. , 73 (1966), 1093-1096

    1966

  9. [9]

    Haldar , Solutions of Fermat-Type Partial Differential–Difference Equations in Cn, Mediterranean J

    G. Haldar , Solutions of Fermat-Type Partial Differential–Difference Equations in Cn, Mediterranean J. Math., 20(1) (2023), 50

    2023

  10. [10]

    P. C. Hu and C. C. Yang , Factorization of holomorphic mappings, Complex variables , 27 (1995), 235- 244

    1995

  11. [11]

    P. C. Hu and C. C. Yang , Uniqueness of meromorphic functions on Cm, Complex variables , 30 (1996), 235-270

    1996

  12. [12]

    P. C. Hu , P. Li and C. C. Yang , Unicity of Meromorphic Mappings. Springer, New York (2003 )

    2003

  13. [13]

    P. C. Hu and C. C. Yang , The Tumura-Clunie theorem in several complex variables, Bull. Aust. Math. Soc., 90 (2014), 444-456

    2014

  14. [14]

    R. J. Korhonen , A difference Picard theorem for meromorphic functions of se veral variables, Comput. Methods Funct. Theory , 12 (1) (2012), 343-361

    2012

  15. [15]

    B. Q. Li , On entire solutions of Fermat type partial differential equ ations, Int. J. Math. , 15 (2004), 473-485

    2004

  16. [16]

    B. Q. Li , On meromorphic solutions of f 2 + g2 = 1, Math. Z. , 258 (4) (2008), 763-771

    2008

  17. [17]

    K. Liu , T. B. Cao and H. Z. Cao , Entire solutions of Fermat type differential-difference eq uations, Arch. Math., 99 (2012), 147-155

    2012

  18. [18]

    Liu and L

    K. Liu and L. Z. Yang , On entire solutions of some differential-difference equati ons, Comput. Methods Funct. Theory, 13 (3) (2013), 433-447

    2013

  19. [19]

    L ¨u and W

    F. L ¨u and W. Bi , On entire solutions of certain partial differential equati ons, J. Math. Anal. Appl. , 516 (1) (2022), 126476

    2022

  20. [20]

    L ¨u, On meromorphic solutions of certain partial differential e quations, Canadian Math

    F. L ¨u, On meromorphic solutions of certain partial differential e quations, Canadian Math. Bull. , 2025:1- 15, doi:10.4153/S0008439525000347

    work page doi:10.4153/s0008439525000347 2025

  21. [21]

    Majumder , The Clunie-Hayman theorem in Cm and normality criteria concerning partial derivative, Complex Var

    S. Majumder , The Clunie-Hayman theorem in Cm and normality criteria concerning partial derivative, Complex Var. Elliptic Equ. DOI: 10.1080/17476933.2026.2638291

    work page doi:10.1080/17476933.2026.2638291 2026

  22. [22]

    Majumder and P

    S. Majumder and P. Das , Periodic behavior of meromorphic functions sharing value s with their shifts in several complex variables, Indian J. Pure Appl. Math. https://doi.org/10.1007/s13226-025-00778-6

    work page doi:10.1007/s13226-025-00778-6

  23. [23]

    Majumder , P

    S. Majumder , P. Das and D. Pramanik , Sufficient condition for entire solution of a certain type of partial differential equation in Cm, J. Contemp. Math. Anal. , 60 (5) (2025), 378-395

    2025

  24. [24]

    Majumder and N

    S. Majumder and N. Sarkar , Periodic behavior of meromorphic functions sharing val- ues with their difference operators in several complex varia bles, Indian J. Pure Appl. Math. https://doi.org/10.1007/s13226-025-00916-0

    work page doi:10.1007/s13226-025-00916-0

  25. [25]

    Majumder and N

    S. Majumder and N. Sarkar , Bergweiler-Langley lemmas in several complex variables, Bull. Belgian Math. Soc. , 33 (2026), 190-211

    2026

  26. [26]

    Majumder and N

    S. Majumder and N. Sarkar , Meromorphic functions in several complex variables satis fying partial derivative conditions, Iran J. Sci. , DOI: 10.1007/s40995-026-01986-3

    work page doi:10.1007/s40995-026-01986-3

  27. [27]

    Majumder , N

    S. Majumder , N. Sarkar and D. Pramanik , Solutions of complex Fermat-type difference equations in several variables, Houston J. math., 51 (3) (2025), 453-482

    2025

  28. [28]

    Montel , Lecons sur les familles normales de fonctions analytiques et leurs applications, Gauthier- Villars, Paris (1927), 135-136

    P. Montel , Lecons sur les familles normales de fonctions analytiques et leurs applications, Gauthier- Villars, Paris (1927), 135-136

    1927

  29. [29]

    E. G. Saleeby , Entire and meromorphic solutions of Fermat type partial di fferential equations, Analysis, 19 (1999), 369-376

    1999

  30. [30]

    E. G. Saleeby , On complex analytic solutions of certain trinomial functi onal and partial differential equations, Aequat. Math., 85 (2013), 553-562

    2013

  31. [31]

    Stoll , Value distribution on parabolic spaces, Lecture Notes in M ath., 600 (1977), Springer-Verlag

    W. Stoll , Value distribution on parabolic spaces, Lecture Notes in M ath., 600 (1977), Springer-Verlag

    1977

  32. [32]

    Taylor and A

    R. Taylor and A. Wiles , Ring-theoretic properties of certain Hecke algebra, Ann. Math. , 141 (1995), 553-572

    1995

  33. [33]

    Wiles , Modular elliptic curves and Fermats last theorem, Ann

    A. Wiles , Modular elliptic curves and Fermats last theorem, Ann. Math. , 141 (1995), 443–551

    1995

  34. [34]

    Xu and T

    L. Xu and T. B. Cao , Solutions of complex Fermat-Type partial difference and di fferential-difference equations, Mediterr. J. Math. , 15 (2018), 227

    2018

  35. [35]

    Xu and T

    L. Xu and T. B. Cao , Correction to: Solutions of Complex Fermat-Type Partial D ifference and Differential-Difference Equations, Mediterr. J. Math. , 17 (2020), 1-4

    2020

  36. [36]

    H. Y. Xu , S. Liu and Q. P. Li , Entire solutions for several systems of nonlinear differen ce and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl. , 483 (2) (2020), 123641

    2020

  37. [37]

    H. Y. Xu and H. W ang, Notes on the existence of entire solutions for several part ial differential-difference equations, Bull. Iran. Math. Soc. , 47 (2020), 1477-1489

    2020

  38. [38]

    H. Y. Xu and H. M. Srivastava , A study of transcendental entire solutions of several nonlinear partial differential equations, Proc. Edinb. Math. Soc. Published online 2025:1-41. doi:10.1017/S0013091525100825. 30 S. Majumder, J. Banerjee and A. Banerjee 1∗Department of Mathematics, Raiganj University, Raiganj, W est Bengal-733134, India. Email address : s...

    work page doi:10.1017/s0013091525100825 2025