$\mathbb{C}^n$中二次三项式型偏微分–差分方程的整解
Entire Solutions for quadratic trinomial-type partial differential-difference equations in $ \mathbb{C}^n $
AI总结：
本文借助Nevanlinna理论刻画$\mathbb{C}^n$中两类含平移或差分的二次三项式型偏微分–差分方程整解的存在性与形式，推广并改进了已有结果。
摘要：
本文利用Nevanlinna理论，研究$\mathbb{C}^n$中二次三项式型偏微分–差分方程整解$f$的存在性及其形式：所研究的方程为$a\left(α\dfrac{\partial f(z)}{\partial z_i}+β\dfrac{\partial f(z)}{\partial z_j}\right)^2+2ω\left(α\dfrac{\partial f(z)}{\partial z_i}+β\dfrac{\partial f(z)}{\partial z_j}\right)f(z+c)+bf(z+c)^2=e^{g(z)}$，以及$a\left(α\dfrac{\partial f(z)}{\partial z_i}+β\dfrac{\partial f(z)}{\partial z_j}\right)^2+2ω\left(α\dfrac{\partial f(z)}{\partial z_i}+β\dfrac{\partial f(z)}{\partial z_j}\right)Δ_cf(z)+b[Δ_cf(z)]^2=e^{g(z)}$，其中$a,ω,b\in\mathbb{C}$，$g$是$\mathbb{C}^n$中的多项式，且$Δ_cf(z)=f(z+c)-f(z)$。本文的主要结果改进了整数$n\geq2$且$1\leq i<j\leq n$时$\mathbb{C}^n$中的若干存在性结果；其推论推广了Xu等人关于$\mathbb{C}^2$中任意系数三项式方程的结果。此外，文中给出了若干例子以验证主要结果的结论。
英文摘要：
In this paper, utilizing Nevanlinna theory, we study existence and forms of the entire solutions $ f $ of the quadratic trinomial-type partial differential-difference equations in $ \mathbb{C}^n $ \begin{align*} a\left(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right)^2 + 2 ω\left(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right) f(z + c) + b f(z + c)^2 = e^{g(z)} \end{align*} and \begin{align*} a\left(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right)^2 & + 2 ω\left(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right) Δ_cf(z) + b [Δ_cf(z)]^2 = e^{g(z)}, \end{align*} where $ a, ω, b\in\mathbb{C} $, $ g $ is a polynomial in $ \mathbb{C}^n $ and $ Δ_cf(z)=f(z+c)-f(z) $. The main results of the paper improve several existence results in $ \mathbb{C}^n $ for integer $ n\geq 2 $ and $ 1\leq i<j\leq n $ and their corollaries of the paper are an extension of the results of Xu \emph{et al. } for trinomial equation with arbitrary coefficient in $ \mathbb{C}^2 $. Moreover, examples are exhibited to validate the conclusion of the main results.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
