Advertisement

The Ramanujan Journal

, Volume 15, Issue 3, pp 415–433 | Cite as

Meromorphic solutions of equations over non-Archimedean fields

  • Ta Thi Hoai An
  • Alain EscassutEmail author
Article

Abstract

In this paper, we give some conditions to assure that the equation P(X)=Q(Y) has no meromorphic solutions in all K, where P and Q are polynomials over an algebraically closed field K of characteristic zero, complete with respect to a non-Archimedean valuation. In particular, if P and Q satisfy the hypothesis (F) introduced by H. Fujimoto, a necessary and sufficient condition is obtained when deg P=deg Q. The results are presented in terms of parametrization of a projective curve by three entire functions. In this way we also obtain similar results for unbounded analytic functions inside an open disk.

Keywords

Nevanlinna theory Functional equations Uniqueness polynomials Meromorphic functions Curve Singularity 

Mathematics Subject Classification (2000)

12E05 11S80 30D35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    An, T.T.H., Wang, J.T.-Y., Wong, P.-M.: Unique range sets and uniqueness polynomials in positive characteristic. Acta Arith. 109(3), 259–280 (2003) zbMATHMathSciNetGoogle Scholar
  2. 2.
    An, T.T.H., Wang, J.T.-Y., Wong, P.-M.: Unique range sets and uniqueness polynomials in positive characteristic II. Acta Arith. 116(2), 115–143 (2005) zbMATHMathSciNetGoogle Scholar
  3. 3.
    An, T.T.H., Wang, J.T.-Y.: A note on uniqueness polynomials of entire functions. Preprint Google Scholar
  4. 4.
    Berkovich, V.: Spectral Theory and Analytic Geometry over Non-Archimedean Fields. Mathematical Survey and Monographs, vol. 33. American Mathematical Society, Providence (1990) zbMATHGoogle Scholar
  5. 5.
    Boutabaa, A.: Théorie de Nevanlinna p-adique. Manuscripta Math. 67, 250–269 (1990) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Boutabaa, A., Escassut, A.: An improvement of the p-adic Nevanlinna theory and application to the meromorphic functions. In: Kakol, J., et al.(eds.) p-Adic Functional Analysis. Proceedings of the 5th International Conference in Poznan, Poland, June 1–5, 1998. Lect. Notes Pure Appl. Math., vol. 207, pp. 29–38. Marcel Dekker, New York (1999) Google Scholar
  7. 7.
    Boutabaa, A., Cherry, W., Escassut, A.: Unique range sets in positive characteristic. Acta Arith. 103, 169–189 (2002) zbMATHMathSciNetGoogle Scholar
  8. 8.
    Cherry, W., Ru, M.: Rigid analytic Picard theorems. Am. J. Math. 126, 873–889 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cherry, W., Wang, J.T.-Y.: Non-Archimedean analytic maps to algebraic curves. In: Cherry, W., Yang, C.C. (eds.) Value Distribution Theory and Complex Dynamics. Contemporary Mathematics, vol. 303, pp. 7–36. American Mathematical Society, Providence (2002) Google Scholar
  10. 10.
    Cherry, W., Ye, Z.: Non-Archimedean Nevanlinna theory in several variables and the non-Archimedean Nevanlinna inverse problem. Trans. Am. Math. Soc. 349(12), 5043–5071 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Escassut, A.: Analytic Elements in p-adic Analysis. World Scientific, Singapore (1995) zbMATHGoogle Scholar
  12. 12.
    Escassut, A., Yang, C.C.: The functional equation P(f)=Q(g) in a p-adic field. J. Number Theory 105(2), 344–360 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Fujimoto, H.: On uniqueness of meromorphic functions sharing finite sets. Am. J. Math. 122, 1175–1203 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Fulton, W.: Algebraic Curves. Benjamin, New York (1969) zbMATHGoogle Scholar
  15. 15.
    Green, M.L.: Some Picard theorems for holomorphic maps to algebraic varieties. Am. J. Math. 97, 43–75 (1975) zbMATHCrossRefGoogle Scholar
  16. 16.
    Khoai, H.H., An, T.T.H.: On uniquenes polynomials and bi-urs for p-adic meromorphic functions. J. Number Theory 87, 211–221 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Khoai, H.H., Tu, M.V.: Nevanlinna-Cartan theorem. Int. J. Math. 6(5), 719–731 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Khoai, H.H., Yang, C.C.: On the functional equation P(f)=Q(g). In: Value Distribution Theory, pp. 201–231. Marcel Dekker, New York (2003) Google Scholar
  19. 19.
    Mayerhofer, E.: Rational decompositions of p-adic meromorphic functions. SCMJ 61(1), 1–13 (2005) zbMATHMathSciNetGoogle Scholar
  20. 20.
    Wang, J.T.-Y.: Uniqueness polynomials and bi-unique range sets for rational functions and non-Archimedean meromorphic functions. Acta Arith. 104, 183–200 (2002) zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Yang, C.C., Li, P.: Some further results on the functional equation P(f)=Q(g). In: Value Distribution Theory and Related Topics, pp. 219–231. Kluwer, Boston (2004) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institute of MathematicsHanoiViet Nam
  2. 2.Laboratoire de Mathématiques, UMR 6620Université Blaise Pascal (Clermont-Ferrand)Aubiere-CedexFrance

Personalised recommendations