Abstract
In this paper, we prove some difference analogue of second main theorems of meromorphic mapping from ℂm into an algebraic variety V intersecting a finite set of fixed hypersurfaces in subgeneral position. As an application, we prove a result on algebraic degeneracy of holomorphic curves on \({\cal P}_c^1\) intersecting hypersurfaces and difference analogue of Picard’s theorem on holomorphic curves. Furthermore, we obtain a second main theorem of meromorphic mappings intersecting hypersurfaces in N-subgeneral position for Veronese embedding in ℙn(ℂ) and a uniqueness theorem sharing hypersurfaces. Our second main theorem and difference analogue of Picard’s theorem recover the results of Cao-Korhonen [1] and Halburd-Korhonen-Tohge [8], respectively. By a way, we also obtain uniqueness theorems of meromorphic mappings which improve the result of Dulock-Ru [4].
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References
T. B. Cao and R. Korhonen, A new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variable, J. Math. Anal. Appl., 444 (2016), 1114–1132.
T. B. Cao and R. Korhonen, Value distribution theory of q-differences in several complex variables, Anal. Math., 46 (2020), 699–736.
T. B. Cao and L. Xu, Logarithmic difference lemma in several complex variables and partial difference equations, Ann. Mat. Pura Appl., 199 (2020), 767–794.
M. Dulock and M. Ru, A uniqueness theorem for holomorphic curves into encountering hypersurfaces in projective space, Complex Var. Elliptic Equ., 53 (2008), 797–802.
J. H. Evertse and R. G. Ferretti, A generalization of the Subspace Theorem with Polynomials of Higher Degree, Developments in Mathematics, 16, Springer-Verlag (New York, 2008), pp. 175–198.
R. Halburd and R. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl., 314 (2006), 477–487.
R. Halburd and R. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math., 31 (2006), 463–478.
R. Halburd, R. Korhonen and K. Tohge, Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc., 366 (2014), 4267–4298.
R. Korhonen, N. Li and K. Tohge, Difference analogue of Cartan second main theorem for slowly moving periodic targets, Ann. Acad. Fenn. Math., 41 (2016), 523–549.
E. I. Nochka, On the theory of meromorphic curves, Soviet Math. Dokl., 27 (1983) 377–381.
J. Noguchi, A note on entire pseudo-holomorphic curves and the proof of Cartan-Nochka’s theorem, Kodai Math. J., 28 (2005), 336–346.
L. T. Tuyet, N. D. Tuyen and P. D. Thoan, Second main theorem and uniqueness problem of zero-order meromorphic mappings for hyperplanes in subgeneral position, Bull. Korean. Math. Soc., 55 (2018), 205–226.
H. T. Phuong, On unique range sets for holomorphic maps sharing hypersurfaces without counting multiplicity, Acta. Math. Vietnam., 34 (2009), 351–360.
S. D. Quang, Degeneracy second main theorems for meromorphic mappings into projective varieties with hypersurfaces, Trans. Amer. Math. Soc., 371 (2019), 2431–2453.
M. Ru, Holomorphic curves into algebraic varieties, Ann. of Math. (2), 169 (2009), 255–267.
M. Sombra, Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz, Algorithms for algebra (Eindhoven, 1996), J. Pure Appl. Algebra, 117/118 (1997), 565–599.
V. D. Waerden, Algebra, vol. 2, 7th ed., Spinger-Verlag (New York, 1991).
P. M. Wong, H. F. Law and P. P. W. Wong, A second main theorem on ℙn(ℂ)for difference operator, Sci. China Ser. A, 52 (2009), 2751–2758.
J. Zheng and R. Korhonen, Studies of differences from the point of view of Nevanlinna theory, Trans. Amer. Math. Soc., 373 (2020), 4285–4318.
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The authors express thanks to the referees and the editorial board for reading the manuscript very carefully and making some valuable suggestions and comments towards the improvement of the paper.
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Pei-Chu Hu is supported by NSFC of Shandong (No. ZR2018MA014), PCSIRT (No. IRT1264) and The Fundamental Research Funds of Shandong University (No. 2017JC019).
Nguyen Van Thin is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 01/2020/STS01.
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Hu, PC., Thin, N.V. Difference analogue of second main theorems for meromorphic mapping into algebraic variety. Anal Math 47, 811–842 (2021). https://doi.org/10.1007/s10476-021-0089-3
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DOI: https://doi.org/10.1007/s10476-021-0089-3