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Entire solutions of Fermat-Type Partial Differential-Difference equations in higher dimensions
Abstract
The purpose of this paper is to investigate the finite-order entire solutions of the following Fermat-type partial differential-difference equations: (P1(z)f(z+c))m1+(P2(z)f(z)+sidesetj=1m∑ Qj(z)∂zj(f(z)))m2=P(z), (P1(z)f(z))m1+(P2(z)f(z+c)+sidesetj=1m∑ Qj(z)∂zj(f(z)))m2=P(z) and (sidesetj=1m∑ Qj(z)∂zj(f(z)))m1+(P1(z)f(z+c)+P2(z)f(z))m2=P(z), in ℂm, where m1,m2∈ℕ such that m1+m2>2, c=(c1,c2,…,cm)∈ℂm, P(not≡ 0), P1, P2 and Qi (i=1,2,…,m) are polynomials in ℂm such that P(z) is irreducible in ℂm. The results established in this paper generalize those of Xu et al. \cite{Xu-Luo-Cao-2025} from ℂ to ℂm.
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BibTeX
@misc{majumder2026entire,
title = {{Entire solutions of Fermat-Type Partial Differential-Difference equations in higher dimensions}},
author = {Sujoy Majumder and Jhilik Banerjee and Abhijit Banerjee and Elias Saleeby},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01931},
eprint = {2611.01931},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01931},
}
RIS
TY - UNPB
TI - Entire solutions of Fermat-Type Partial Differential-Difference equations in higher dimensions
AU - Sujoy Majumder
AU - Jhilik Banerjee
AU - Abhijit Banerjee
AU - Elias Saleeby
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01931
UR - https://arxiv.org/abs/2611.01931
AB - The purpose of this paper is to investigate the finite-order entire solutions of the following Fermat-type partial differential-difference equations: (P_1(z)f(z+c))^m_1+(P_2(z)f(z)+sideset_j=1^m∑ Q_j(z)∂_z_j(f(z)))^m_2=P(z), (P_1(z)f(z))^m_1+(P_2(z)f(z+c)+sideset_j=1^m∑ Q_j(z)∂_z_j(f(z)))^m_2=P(z) and (sideset_j=1^m∑ Q_j(z)∂_z_j(f(z)))^m_1+(P_1(z)f(z+c)+P_2(z)f(z))^m_2=P(z), in ℂ^m, where m_1,m_2∈ℕ such that m_1+m_2>2, c=(c_1,c_2,…,c_m)∈ℂ^m, P(not≡ 0), P_1, P_2 and Q_i (i=1,2,…,m) are polynomials in ℂ^m such that P(z) is irreducible in ℂ^m. The results established in this paper generalize those of Xu et al. \cite{Xu-Luo-Cao-2025} from ℂ to ℂ^m.
ER -