Abstract
With the aid of Gel'fond's method [1, 2], which makes it possible to show that there exist at least two algebraically independent quantities among several values of the exponential function, and by using certain additional considerations, the author obtains a result concerning the algebraic independence of the values of the exponential and the elliptic functions.
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A. O. Gel'fond, “On algebraic independence of certain classes of transcendental numbers,” Uspekhi Matem. Nauk,4, No. 5, 14–48 (1949) (see also A. O. Gel'fond, Selected Works [in Russian], Nauka, Moscow (1973), pp. 191–222).
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A. A. Shmelev, “On the algebraic independence of exponentials,” Matem. Zametki,17, No. 3, 407–418 (1975).
N. I. Fel'dman, “On joint approximations of periods of an elliptic function by algebraic numbers,” Izv. AN SSSR, Ser. Matem.,22, 563–576 (1958).
A. A. Shmelev, “On approximating the roots of certain transcendental equations,” Matem. Zametki,7, No. 2, 203–210 (1970).
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Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 195–202, August, 1976.
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Shmelev, A.A. Algebraic independence of values of exponential and elliptic functions. Mathematical Notes of the Academy of Sciences of the USSR 20, 669–673 (1976). https://doi.org/10.1007/BF01155871
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DOI: https://doi.org/10.1007/BF01155871