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The values of exponential polynomials at algebraic points. I


Author: Carlos Julio Moreno
Journal: Trans. Amer. Math. Soc. 186 (1973), 17-31
MSC: Primary 10F35
DOI: https://doi.org/10.1090/S0002-9947-1973-0325545-2
MathSciNet review: 0325545
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Abstract: A strengthening of Siegel's proof of the Hermite-Lindemann Theorem is given. The results are used to obtain lower bounds for the values of exponential polynomials at algebraic points. The question of how well the root of an exponential polynomial can be approximated by algebraic numbers is considered, and a lower bound is obtained for the absolute value of the difference between a root of the exponential polynomial and an algebraic number.


References [Enhancements On Off] (What's this?)

  • [1] N. I. Fel'dman, Arithmetic properties of the solutions of a transcendental equation, Vestnik Moskov. Univ. Ser. I Mat. Meh. 1964, no. 1, 13-20; English transl., Amer. Math. Soc. Transl. (2) 66 (1968), 145-153. MR 28 #2091. MR 0158869 (28:2091)
  • [2] A. Gel'fond, Transcendental and algebraic numbers, GITTL, Moscow, 1952; English transl., Dover, New York, 1960. MR 15, 292; 22 #2598. MR 0111736 (22:2598)
  • [3] C. Hermite, Sur la fonction exponentielle, Acad. Sci. Paris 77 (1873); Ouvres. III, 150-181.
  • [4] S. Lang, Introduction to transcendental numbers, Addison-Wesley, Reading, Mass., 1966. MR 35 # 5397. MR 0214547 (35:5397)
  • [5] F. Lindemann, Über die Zahl $ \pi $, Math. Ann. 20 (1882).
  • [6] K. Mahler, Zur Approximation der Exponential Funktion und des Logarithmus, J. Reine Angew. Math. 166 (1931/32).
  • [7] D. Mardohai-Boltouski, On some properties of transcendental numbers of the first class, Mat. Sb. 34 (1927).
  • [8] C. J. Moreno, The values of exponential polynomials at algebraic points. II (to appear). MR 0347746 (50:247)
  • [9] A. Šidlovskiĭ, On the transcendence of the values of a class of entire functions satisfying linear differential equations, Dokl. Akad. Nauk SSSR 105 (1955), 35-37. (Russian) MR 17, 947. MR 0076806 (17:947c)
  • [10] C. Siegel, Über einige Anwendungen Diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1929/30, No. 1.
  • [11] -, Transcendental numbers, Ann. of Math. Studies, no. 16, Princeton Univ. Press, Princeton, N.J., 1949. MR 11, 330. MR 0032684 (11:330c)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0325545-2
Keywords: Algebraic, diophantine, approximation, exponential polynomial, Hermite-Lindemann Theorem
Article copyright: © Copyright 1973 American Mathematical Society

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