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PreprintarXiv licencearXiv

Finite-Order Entire Solutions to Fermat-Type Partial Differential-Difference Systems in ℂn

  • Sujoy Majumder
  • Jhilik Banerjee
  • Abhijit Banerjee
  • Elias G. Saleeby
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Abstract

The primary objective of this paper is to determine the existence and explicit form of finite-order entire solutions in ℂn of the following system of Fermat-type partial differential-difference equations: ((∂ f1(z))/(∂ z1))n1 + (f2 (z+c)−f1(z) )m1= 1, medskip ((∂ f2(z))/(∂ z1))n2 + (f1 (z+c )−f2(z) )m2= 1, for several choices of the positive integers n1, n2, m1, and m2, where c=(c1,c2,…,cn). We obtain structural classifications and nonexistence results in the exponent regimes specified in the main theorems, extending results of Xu et al. \cite{XLL1} from ℂ2 to ℂn. Several examples illustrate the resulting solution families.

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Cite

BibTeXRIS
BibTeX
@misc{majumder2026finiteorder,
  title = {{Finite-Order Entire Solutions to Fermat-Type Partial Differential-Difference Systems in $\mathbbC^n$}},
  author = {Sujoy Majumder and Jhilik Banerjee and Abhijit Banerjee and Elias G. Saleeby},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.05240},
  eprint = {2606.05240},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.05240},
}
RIS
TY  - UNPB
TI  - Finite-Order Entire Solutions to Fermat-Type Partial Differential-Difference Systems in ℂ^n
AU  - Sujoy Majumder
AU  - Jhilik Banerjee
AU  - Abhijit Banerjee
AU  - Elias G. Saleeby
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2606.05240
UR  - https://arxiv.org/abs/2606.05240
AB  - The primary objective of this paper is to determine the existence and explicit form of finite-order entire solutions in ℂ^n of the following system of Fermat-type partial differential-difference equations: ((∂ f_1(z))/(∂ z_1))^n_1 + (f_2 (z+c)−f_1(z) )^m_1= 1, medskip ((∂ f_2(z))/(∂ z_1))^n_2 + (f_1 (z+c )−f_2(z) )^m_2= 1, for several choices of the positive integers n_1, n_2, m_1, and m_2, where c=(c_1,c_2,…,c_n). We obtain structural classifications and nonexistence results in the exponent regimes specified in the main theorems, extending results of Xu et al. \cite{XLL1} from ℂ^2 to ℂ^n. Several examples illustrate the resulting solution families.
ER  -