On the existence of entire solutions to a system of nonlinear Fermat-type partial differential-difference equations
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908686381744128 |
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| author | Xu, Junfeng Majumder, Sujoy Pramanik, Debabrata |
| author_facet | Xu, Junfeng Majumder, Sujoy Pramanik, Debabrata |
| contents | The aim of this study is to investigate the precise form of finite-order entire solutions to the following system of Fermat-type partial differential-difference equations: \beas \begin{cases} \left(\frac{\partial f_1\left(z_1, z_2, \ldots, z_m \right)}{\partial z_1}\right)^{n_1} + f_2^{m_1} \left(z_1 + c_1, z_2 + c_2, \ldots, z_m + c_m \right) = 1,\\ \left(\frac{\partial f_2\left(z_1, z_2, \ldots, z_m \right)}{\partial z_1}\right)^{n_2} + f_1^{m_2} \left(z_1 + c_1, z_2 + c_2, \ldots, z_m + c_m \right) = 1 \end{cases} \eeas
for various combinations of the positive integers $n_1$, $n_2$, $m_1$ and $m_2$. Our results extend the work of Xu et al. (Entire solutions for several systems of non-linear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl., 483(2), 2020), generalizing the setting $\mathbb{C}^2$ to $\mathbb{C}^m$. Several examples are provided to illustrate the applicability and sharpness of the obtained results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_02040 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the existence of entire solutions to a system of nonlinear Fermat-type partial differential-difference equations Xu, Junfeng Majumder, Sujoy Pramanik, Debabrata Complex Variables 39A45, 32H30, 39A14 and 35A20 The aim of this study is to investigate the precise form of finite-order entire solutions to the following system of Fermat-type partial differential-difference equations: \beas \begin{cases} \left(\frac{\partial f_1\left(z_1, z_2, \ldots, z_m \right)}{\partial z_1}\right)^{n_1} + f_2^{m_1} \left(z_1 + c_1, z_2 + c_2, \ldots, z_m + c_m \right) = 1,\\ \left(\frac{\partial f_2\left(z_1, z_2, \ldots, z_m \right)}{\partial z_1}\right)^{n_2} + f_1^{m_2} \left(z_1 + c_1, z_2 + c_2, \ldots, z_m + c_m \right) = 1 \end{cases} \eeas for various combinations of the positive integers $n_1$, $n_2$, $m_1$ and $m_2$. Our results extend the work of Xu et al. (Entire solutions for several systems of non-linear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl., 483(2), 2020), generalizing the setting $\mathbb{C}^2$ to $\mathbb{C}^m$. Several examples are provided to illustrate the applicability and sharpness of the obtained results. |
| title | On the existence of entire solutions to a system of nonlinear Fermat-type partial differential-difference equations |
| topic | Complex Variables 39A45, 32H30, 39A14 and 35A20 |
| url | https://arxiv.org/abs/2512.02040 |