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Siegel-Shidlovsky method in p-adic domain

Abstract

This paper presents a review of the arithmetic properties of F-series.

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References

  1. 1.

    Y. André, G-Functions and Geometry, Max-Planck-Institut für Mathematik, Bonn. Aspects of Math., Vol. 13, Friedr. Vieweg & Sohn, Wiesbaden (1989).

    Google Scholar 

  2. 2.

    Y. André, “Séries Gevrey de type arithméique. II: Transcendance sans transcendance,” Ann. Math., 151, 741–756 (2000).

    MATH  Article  Google Scholar 

  3. 3.

    D. Bertrand, “On André’s proof of the Siegel-Shidlovsky theorem,” Publ. Keio Univ., 27, 51–63 (1999).

    MathSciNet  Google Scholar 

  4. 4.

    D. Bertrand and F. Beukers, “Equations différentielles linéaires et majorations de multiplicités,” Ann. Sci. École Norm. Sup., 18, 181–192 (1985).

    MATH  MathSciNet  Google Scholar 

  5. 5.

    D. Bertrand, V. Chirskii, and J. Yebbou, “Effective estimates for global relations on Euler-type series,” Ann. Fac. Sci. Toulouse Math., 13, No. 2, 241–260 (2004).

    MATH  MathSciNet  Google Scholar 

  6. 6.

    E. Bombieri, “On G-functions,” in: Recent Progress in Analytic Number Theory, Symp. Durham 1979, Vol. 2, Academic Press, London (1981), pp. 1–67.

    Google Scholar 

  7. 7.

    V. G. Chirskii, “On non-trivial global relations,” Moscow Univ. Math. Bull., 44, No. 5, 41–44 (1989).

    MathSciNet  Google Scholar 

  8. 8.

    V. G. Chirskii, “Global relations,” Math. Notes, 48, No. 1–2, 795–798 (1990).

    MathSciNet  Google Scholar 

  9. 9.

    G. V. Chudnovsky, “On applications of Diophantine approximations,” Proc. Nat. Acad. Sci. U.S.A., 81, 7261–7265 (1985).

    Article  MathSciNet  Google Scholar 

  10. 10.

    Yu. Flicker, “On p-adic G-functions,” J. London Math. Soc., 15, No. 3, 395–402 (1977).

    MATH  Article  MathSciNet  Google Scholar 

  11. 11.

    A. I. Galochkin, “On the algebraic independence of values of E-functions at certain transcendental points,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 58–63 (1970).

  12. 12.

    E. M. Matveev, “Linear forms of the values of G-functions, and Diophantine equations,” Mat. Sb. (N. S.), 117(159), No. 3, 379–396 (1982).

  13. 13.

    V. Kh. Salikhov, “Formal solutions of linear differential equations and their applications in the theory of transcendental numbers,” Tr. Mosk. Mat. Obshch., 51, 223–256 (1988).

    MATH  Google Scholar 

  14. 14.

    V. Kh. Salikhov, “Irreducibility of hypergeometric equations and algebraic independence of values of E-functions” [in Russian], Acta Arith., 53, 453–471 (1990).

    MATH  MathSciNet  Google Scholar 

  15. 15.

    A. B. Shidlovskii, Transcendental Numbers, Walter de Gruyter, Berlin (1989).

    MATH  Google Scholar 

  16. 16.

    C. L. Siegel, “Über einige Anwendungen Diophantischer Approximationen,” Abh. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 1, 1–70 (1929).

  17. 17.

    J. Yebbou, “Calcul de facteurs déterminants,” J. Differential Equations, 72, 140–148 (1988).

    MATH  Article  MathSciNet  Google Scholar 

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Correspondence to V. G. Chirskii.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 221–230, 2005.

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Chirskii, V.G. Siegel-Shidlovsky method in p-adic domain. J Math Sci 146, 5791–5797 (2007). https://doi.org/10.1007/s10958-007-0393-x

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Keywords

  • Algebraic Number
  • Diophantine Equation
  • Monic Polynomial
  • Arithmetic Property
  • Global Relation