Abstract
We consider some arithmetic properties of values of meromorphic functions \(g_1(z)\), …, \(g_m(z)\) such that each of \(g'_i(z)\) is algebraically dependent over a field \(K\) of algebraic numbers, \([K:\mathbb Q]<\infty\), with the functions \(g_1(z),\dots,g_m(z)\). We show that if all \(\{g_i(z)\}\) are meromorphic of finite order, then either they all are rational functions, or they all are rational functions of some exponential, or they all are elliptic functions, or there exists a discrete set \(U\) such that the number of points \(z\notin U\) such that all \(\{g_i( z)\}\) lie in \(K\) is finite.
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Acknowledgments
The authors wish to express gratitude to the referee for a number of useful remarks that have made for the improvement of the paper.
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Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 863–875 https://doi.org/10.4213/mzm13431.
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Podkopaeva, V.A., Yanchenko, A.Y. On a Refinement of the Schneider–Lang theorem. Math Notes 113, 804–814 (2023). https://doi.org/10.1134/S000143462305022X
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DOI: https://doi.org/10.1134/S000143462305022X