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Register a new userUnicity on entire function concerning its differential polynomials in Several complex variables
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XiaoHuang Huang
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In this paper, we study the uniqueness of the differential polynomials of entire functions. We prove the following result: Let $f(z)$ be a nonconstant entire function on $mathbb{C}^{n}$ and $g(z)=b_{-1}+sum_{i=0}^{n}b_{i}D^{k_{i}}f(z)$, where $b_{-1}$ and $b_{i} (i=0ldots,n)$ are small meromorphic functions of $f$, $k_{i}geq0 (i=0ldots,n)$ are integers. Let $a_{1}(z) otequivinfty, a_{2}(z) otequivinfty$ be two distinct small meromorphic functions of $f(z)$. If $f(z)$ and $g(z)$ share $a_{1}(z)$ CM, and $a_{2}(z)$ IM. Then either $f(z)equiv g(z)$ or $a_{1}=2a_{1}=2$, $$f(z)equiv e^{2p}-2e^{p}+2,$$ and $$g(z)equiv e^{p},$$ where $p(z)$ is a non-constant entire function on $mathbb{C}^{n}$.
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