Second main theorems with weighted counting functions and algebraic dependence of meromorphic mappings

Duc Quang, Si

doi:10.1090/proc/13061

Second main theorems with weighted counting functions and algebraic dependence of meromorphic mappings
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Proc. Amer. Math. Soc. 144 (2016), 4329-4340 Request permission

Abstract:

The purpose of this article is twofold. The first is to prove a new second main theorem for meromorphic mappings and moving hyperplanes of $\mathbb {P}^n(\mathbb {C})$, where the counting functions are truncated multiplicities and have different weights. Our result is an extension of previous second main theorems for moving hyperplanes with the truncated (to level $n$) counting functions. As its application, the second purpose of this article is to prove a new algebraic dependence theorem for meromorphic mappings having the same inverse images of some moving hyperplanes, where the moving hyperplanes involve the assumption with different roles.

References

H. Cartan, Sur les zeros des combinasons lineaires de $p$ fonctions holomorphes donnees, Mathematica 7 (1933), 5-31.
Zhihua Chen, Yanhua Li, and Qiming Yan, Uniqueness problem with truncated multiplicities of meromorphic mappings for moving targets, Acta Math. Sci. Ser. B (Engl. Ed.) 27 (2007), no. 3, 625–634. MR 2339402, DOI 10.1016/S0252-9602(07)60062-9
I. E. Nochka, On the theory of meromorphic functions, Sov. Math. Dokl. 27 (1983), 377-381.
Junjiro Noguchi and Takushiro Ochiai, Geometric function theory in several complex variables, Translations of Mathematical Monographs, vol. 80, American Mathematical Society, Providence, RI, 1990. Translated from the Japanese by Noguchi. MR 1084378, DOI 10.1090/mmono/080
S. D. Quang, Second main theorems for meromorphic mappings intersecting moving hyperplanes with truncated counting functions and unicity problem, Abh. Math. Semin. Univ. Hambg. (2015) pp 1-18, DOI: 10.1007/s12188-015-0114-1.
Si Duc Quang, Algebraic dependences of meromorphic mappings sharing few moving hyperplanes, Ann. Polon. Math. 108 (2013), no. 1, 61–73. MR 3021268, DOI 10.4064/ap108-1-5
Si Duc Quang and Do Phuong An, Unicity of meromorphic mappings sharing few moving hyperplanes, Vietnam J. Math. 41 (2013), no. 4, 383–398. MR 3142403, DOI 10.1007/s10013-013-0035-1
Min Ru, A uniqueness theorem with moving targets without counting multiplicity, Proc. Amer. Math. Soc. 129 (2001), no. 9, 2701–2707. MR 1838794, DOI 10.1090/S0002-9939-01-06040-3
Min Ru and Julie Tzu-Yueh Wang, Truncated second main theorem with moving targets, Trans. Amer. Math. Soc. 356 (2004), no. 2, 557–571. MR 2022710, DOI 10.1090/S0002-9947-03-03453-6
Do Duc Thai and Si Duc Quang, Uniqueness problem with truncated multiplicities of meromorphic mappings in several complex variables for moving targets, Internat. J. Math. 16 (2005), no. 8, 903–939. MR 2168074, DOI 10.1142/S0129167X05003132
Duc Thai Do and Duc Quang Si, Second main theorem with truncated counting function in several complex variables for moving targets, Forum Math. 20 (2008), no. 1, 163–179. MR 2386785, DOI 10.1515/FORUM.2008.007
Duc Thoan Pham and Viet Duc Pham, Algebraic dependences of meromorphic mappings in several complex variables, Ukrainian Math. J. 62 (2010), no. 7, 1073–1089. MR 2888660, DOI 10.1007/s11253-010-0414-7
Pham Duc Thoan, Pham Viet Duc, and Si Duc Quang, Algebraic dependence and unicity theorem with a truncation level to 1 of meromorphic mappings sharing moving targets, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 56(104) (2013), no. 4, 513–526. MR 3157097
Bernard Shiffman, Introduction to the Carlson-Griffiths equidistribution theory, Value distribution theory (Joensuu, 1981) Lecture Notes in Math., vol. 981, Springer, Berlin-New York, 1983, pp. 44–89. MR 699133
Wilhelm Stoll, On the propagation of dependences, Pacific J. Math. 139 (1989), no. 2, 311–337. MR 1011216
Katsutoshi Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192 (2004), no. 2, 225–294. MR 2096455, DOI 10.1007/BF02392741