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Date Published:
27/02/2023
Online Published:
11/11/2022
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Views PDF: 46
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Escassut, A. . (2022). A SURVEY ON A FEW RECENT PAPERS IN 𝙋-ADIC VALUE DISTRIBUTION. Thang Long Journal of Science: Mathematics and Mathematical Sciences, 1(4), 27-56. https://science.thanglong.edu.vn/index.php/volc/article/view/41
A SURVEY ON A FEW RECENT PAPERS IN 𝙋-ADIC VALUE DISTRIBUTION
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Abstract
In this article, we propose to present several recent results: a new proof of the p-adic Hermite-Lindemann Theorem, a new proof of the p-adic Gel'fond-Schneider Theorem, exceptional values of meromorphic functions and derivatives and the p-adic Nevanlinna theory applied to small functions. We first have to recall the definitions of the p-adic logarithm and exponential.
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References
[1] Bezivin, J.-P., Wronskien et equations differentielles p-adiques, Acta Arith., 158, no. 1, 6178 (2013).
[2] Bezivin, J.-P., Boussaf, K. and Escassut, A. A. Zeros of the derivative of a p-adic meromorphic function, Bull. Sci. Math., 136, no. 8, 839847 (2012).
[3] Boussaf, K. Picard values of p-adic meromorphic functions, p-Adic Numbers Ultrametric Anal. Appl., 2, no. 4, 285292 (2010).
[4] Boussaf, K., Ojeda, J. and Escassut, A. Primitives of p-adic meromorphic functions, Contemp. Math., 551, 5156 (2011).
[5] Boutabaa, A. Theorie de Nevanlinna p-adique, Manuscripta Math. 67, p. 251-269 (1990).
[6] Boutabaa, A. and Escassut, A. URS and URSIMS for p-adic meromorphic functions inside a disk, Proc. of the Edinburgh Mathematical Society 44, p. 485-504 (2001).
[7] Escassut, A. p-adic Analytic Functions. World Scientic Publishing Co. Pte. Ltd. Singapore, (2021).
[8] Escassut, A. and Yang, C.C. A short note on two p-adic meromorphic functions sharing a few small ones, Rendiconti del Circolo Matematico di Palermo 70 (2), p. 623-630. [9] Ha H. K. On p-adic meromorphic functions, Duke Mathematical Journal, 50, 695-711 (1983).
[10] Hayman, W. K. Meromorphic Functions. Oxford University Press, (1975)
[11] Hu, P.C. and Yang, C.C. Meromorphic Functions over non-Archimedean Fields, Kluwer Academic Publishers, (2000).
[12] Lazard, M. Les z eros des fonctions analytiques sur un corps valué complet, IHES, Publications Mathématiques no. 14, pp. 47-75 ( 1962).
[13] Malher, K. Ein Beweis der Transzendenz der P-adischen Exponentialfunktion, J. reine angew. Math., 169, pp. 61-66 (1932).
[14] Nevanlinna, R. Le théorème de Picard-Borel et la théorie des fonctions méromorphes. Gauthiers-Villars, Paris (1929).
[15] Shamseddine, K. A brief survey of the study of power series and analytic functions on the Levi-Civita elds, Contemporary Mathematics Volume 596, p. 269-279, (2013). [16] Waldschmidt, M. Nombres transcendants, Lecture Notes in Mathematics, 402, Springer Verlag (1974).
[17] Yamanoi, K. The second main theorem for small functions and related problems Acta Mathematica 192, p. 225-294 (2004).
[2] Bezivin, J.-P., Boussaf, K. and Escassut, A. A. Zeros of the derivative of a p-adic meromorphic function, Bull. Sci. Math., 136, no. 8, 839847 (2012).
[3] Boussaf, K. Picard values of p-adic meromorphic functions, p-Adic Numbers Ultrametric Anal. Appl., 2, no. 4, 285292 (2010).
[4] Boussaf, K., Ojeda, J. and Escassut, A. Primitives of p-adic meromorphic functions, Contemp. Math., 551, 5156 (2011).
[5] Boutabaa, A. Theorie de Nevanlinna p-adique, Manuscripta Math. 67, p. 251-269 (1990).
[6] Boutabaa, A. and Escassut, A. URS and URSIMS for p-adic meromorphic functions inside a disk, Proc. of the Edinburgh Mathematical Society 44, p. 485-504 (2001).
[7] Escassut, A. p-adic Analytic Functions. World Scientic Publishing Co. Pte. Ltd. Singapore, (2021).
[8] Escassut, A. and Yang, C.C. A short note on two p-adic meromorphic functions sharing a few small ones, Rendiconti del Circolo Matematico di Palermo 70 (2), p. 623-630. [9] Ha H. K. On p-adic meromorphic functions, Duke Mathematical Journal, 50, 695-711 (1983).
[10] Hayman, W. K. Meromorphic Functions. Oxford University Press, (1975)
[11] Hu, P.C. and Yang, C.C. Meromorphic Functions over non-Archimedean Fields, Kluwer Academic Publishers, (2000).
[12] Lazard, M. Les z eros des fonctions analytiques sur un corps valué complet, IHES, Publications Mathématiques no. 14, pp. 47-75 ( 1962).
[13] Malher, K. Ein Beweis der Transzendenz der P-adischen Exponentialfunktion, J. reine angew. Math., 169, pp. 61-66 (1932).
[14] Nevanlinna, R. Le théorème de Picard-Borel et la théorie des fonctions méromorphes. Gauthiers-Villars, Paris (1929).
[15] Shamseddine, K. A brief survey of the study of power series and analytic functions on the Levi-Civita elds, Contemporary Mathematics Volume 596, p. 269-279, (2013). [16] Waldschmidt, M. Nombres transcendants, Lecture Notes in Mathematics, 402, Springer Verlag (1974).
[17] Yamanoi, K. The second main theorem for small functions and related problems Acta Mathematica 192, p. 225-294 (2004).