Results on Uniqueness Problem for Meromorphic Mappings Sharing Moving Hyperplanes in General Position Under More General
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
Results on Uniqueness Problem for Meromorphic Mappings Sharing Moving Hyperplanes in General Position Under More General and Weak Conditions∗ Zhixue LIU1
Qingcai ZHANG2
Abstract The aim of the paper is to deal with the algebraic dependence and uniqueness problem for meromorphic mappings by using the new second main theorem with different weights involved the truncated counting functions, and some interesting uniqueness results are obtained under more general and weak conditions where the moving hyperplanes in general position are partly shared by mappings from Cn into PN (C), which can be seen as the improvements of previous well-known results. Keywords Algebraic dependence, Uniqueness problem, Meromorphic mapping, Moving hyperplane 2000 MR Subject Classification 32H30, 32A22, 30D35
1 Introduction and Main Results In 1926, Nevanlinna [9] proved that for two non-constant meromorphic functions f and g on the complex plane C, if they have the same inverse images (ignoring multiplicities) for five distinct values in P1 (C), then f = g. If they have the same inverse images, counted with multiplicities, for four distinct values, then g is a special type of a linear fractional transformation of f . We know that the number five of distinct values in Nevanlinna’s five-value theorem cannot be reduced to four. These results are usually called the five-value theorem and four-value theorem, respectively. Nevanlinna theory for meromorphic mappings of Cn into the complex projective space PN (C) intersecting a finite set of fixed hyperplanes or moving hyperplanes was studied deeply as many years previously due to their important values in complex analysis in several variables, and many interesting results were established, please see [3, 10, 26] for example. Over the last few decades, there have been several generalizations of Nevanlinna’s five-value theorem to the case of meromorphic mappings from Cn into the complex projective space PN (C). Fujimoto [7] generalized the Nevanlinna’s well-known five-value theorem to the case of meromorphic Manuscript received August 9, 2017. Revised April 25, 2018. of Mathematics, Renmin University of China, Beijing 100872, China. E-mail: [email protected] 2 Corresponding author. School of Mathematics, Renmin University of China, Beijing 100872, China. E-mail: [email protected] ∗ This work was supported by the Fund of China Scholarship Council (No. 201806360222) 1 School
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Z. X. Liu and Q. C. Zhang
mappings from Cn into PN (C) and obtained that for two linearly non-degenerate meromorphic mappings f, g of Cn into PN (C), if they have the same inverse images of q (≥ 3N +2) hyperplanes counted with multiplicities located in general position, then f = g. After that, many significant contributions along this line were made to find the smaller “q” (see [5, 20, 23]). In recent years, Chen and Yan [4] considered the case of ignoring the multiplicities and verified that q can be relaxed
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