SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES*

doi:10.1007/s10473-023-0218-0

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (2): 821-838.doi: 10.1007/s10473-023-0218-0

SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES^*

Manli Liu¹, Lingyun Gao^2,†, Shaomei Fang¹

1. Department of Mathematics, South China Agricultural University, Guangzhou 510642, China;
2. Department of Mathematics, Jinan University, Guangzhou 510632, China

收稿日期:2021-10-06 修回日期:2022-03-01 出版日期:2023-03-25 发布日期:2023-04-12
通讯作者: †Lingyun GAO, E-mail: tgaoly@jnu.edu.cn.
作者简介:Manli Liu, E-mail: lml6641@163.com; Shaomei Fang, E-mail: dz90@scau.edu.cn
基金资助:
This work was partially supported by the NSFC (11271227, 11271161), the PCSIRT (IRT1264) and the Fundamental Research Funds of Shandong University (2017JC019).

SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES^*

Manli Liu¹, Lingyun Gao^2,†, Shaomei Fang¹

1. Department of Mathematics, South China Agricultural University, Guangzhou 510642, China;
2. Department of Mathematics, Jinan University, Guangzhou 510632, China

Received:2021-10-06 Revised:2022-03-01 Online:2023-03-25 Published:2023-04-12
Contact: †Lingyun GAO, E-mail: tgaoly@jnu.edu.cn.
About author:Manli Liu, E-mail: lml6641@163.com; Shaomei Fang, E-mail: dz90@scau.edu.cn
Supported by:
This work was partially supported by the NSFC (11271227, 11271161), the PCSIRT (IRT1264) and the Fundamental Research Funds of Shandong University (2017JC019).

摘要/Abstract

摘要： In this paper, we mainly investigate the value distribution of meromorphic functions in $\mathbb{C}^m$ with its partial differential and uniqueness problem on meromorphic functions in $\mathbb{C}^m$ and with its $k$-th total derivative sharing small functions. As an application of the value distribution result, we study the defect relation of a nonconstant solution to the partial differential equation. In particular, we give a connection between the Picard type theorem of Milliox-Hayman and the characterization of entire solutions of a partial differential equation.

关键词: meromorphic function in several variables, Nevanlinna theory, partial differential equation, total derivative

Abstract: In this paper, we mainly investigate the value distribution of meromorphic functions in $\mathbb{C}^m$ with its partial differential and uniqueness problem on meromorphic functions in $\mathbb{C}^m$ and with its $k$-th total derivative sharing small functions. As an application of the value distribution result, we study the defect relation of a nonconstant solution to the partial differential equation. In particular, we give a connection between the Picard type theorem of Milliox-Hayman and the characterization of entire solutions of a partial differential equation.

Key words: meromorphic function in several variables, Nevanlinna theory, partial differential equation, total derivative

Manli Liu, Lingyun Gao, Shaomei Fang. SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES^*[J]. 数学物理学报(英文版), 2023, 43(2): 821-838.

Manli Liu, Lingyun Gao, Shaomei Fang. SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES^*[J]. Acta mathematica scientia,Series B, 2023, 43(2): 821-838.

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SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES^*

SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES^*

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