Advertisement

Complex Analysis and Operator Theory

, Volume 8, Issue 8, pp 1747–1759 | Cite as

Uniqueness Theorems for Holomorphic Curves with Hypersurfaces of Fermat–Waring Type

  • Ha Huy KhoaiEmail author
  • Vu Hoai An
  • Le Quang Ninh
Article
  • 160 Downloads

Abstract

In this paper, we establish uniqueness theorems for holomorphic mappings from \(\mathbb C\) to \(P^N({\mathbb C}) \) for the case where the targets are not hyperplanes, but hypersurfaces of Fermat–Waring type.

Keywords

Holomorphic curves Uniqueness theorems 

Mathematics Subject Classification (1991)

Primary 32H02 Secondary 32H30 

Notes

Acknowledgments

The authors would like to thank the referee for his/her valuable suggestions.

References

  1. 1.
    An, V.H., Duc, T.D.: Uniqueness theorems and uniqueness polynomials for holomorphic curves. In: Complex Variables and Elliptic Equations, Vol. 56, Nos. 1–4, pp. 253–262 (2011) (January–April)Google Scholar
  2. 2.
    Dethloff, G., Tan, T.V., Thai, D.D.: An extension of the Cartan–Nochka second main theorem for hypersurfaces. Int. J. Math. 22, 863–885 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Dethloff, G., Quang, S.D., Tan, T.V.: A uniqueness theorem for meromorphic maps with two families of hyperplanes. Proc. Am. Math. Soc. 140, 189–197 (2012)CrossRefzbMATHGoogle Scholar
  4. 4.
    Fujimoto, H.: Uniqueness problem with truncated multiplicities in value distribution theory. Nagoya Math. J. 152, 131–152 (1998)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Fujimoto, H.: On uniqueness for meromorphic functions sharing finite sets. Am. J. Math. 122, 1175–1203 (2000)Google Scholar
  6. 6.
    Li, P., Yang, C.C.: Meromorphic solutions of functional equation with nonconstant coefficients. Proc. Jpn. Acad. Ser. A Math. Sci. 82(10), 83–186 (2006)Google Scholar
  7. 7.
    Mues, E., Reinders, M.: Meromorphic functions sharing one value and unique range sets. Kodai Math. 18, 515–522 (1995)Google Scholar
  8. 8.
    Smiley, L.: Geometric conditions for unicity of holomorphic curves. Contemp. Math. 25, 149–154 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Thai, D.D., Quang, S.D.: Uniqueness problem with truncated multiplicities of meromorphic mappings in several complex variables for moving targets. Int. J. Math. 10, 903–939 (2005)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Yi, H.X.: On a question of Gross. Sci. China Ser. A 38(1), 8–16 (1995)Google Scholar
  11. 11.
    An, D.P., Quang, S.D., Thai, D.D.: The second main theorem for meromorphic mappings into a complex projective space. Acta Math. Vietnam 38, 187–203 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Shirosaki, M.: A family of polynomials with the uniqueness property for linearly non-degenerate holomorphic mappings. Kodai Math. J. 25, 288–292 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Siu, Y.T., Yeung, S.K.: Defects for ample divisors of Abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees. Am. J. Math. 119, 1139–1172 (1997)Google Scholar
  14. 14.
    Hayman, W.K.: Meromorphic Functions. Clarendon, Oxford (1964)Google Scholar
  15. 15.
    Shirosaki, M.: On polynomials which determine holomorphic mappings. J. Math. Soc. Jpn. 49, 289–298 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Masuda, K., Noguchi, J.: A construction of hyperbolic hypersurface of \( P^N(\mathbb{C})\). Math. Ann. 304, 339–362 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Khoai, H.H.: Some remarks on the genericity of unique range set for meromorphic functions. Sci. China Ser. A Math. 48(suppl), 262–267 (2005)Google Scholar
  18. 18.
    Khoai, H.H., Yang C.C.: On the functional equation \( P(f) = Q(g)\). In: Advances in Complex Analysis and Application, Value Distribution Theory and Related Topics, pp. 201–2008. Kluwer Academic Publishers, Boston, MA (2004)Google Scholar
  19. 19.
    Li, P., Yang, C.C.: On the unique range sets of meromorphic functions. Proc. Am. Math. Soc. 124, 177–195 (1996)CrossRefzbMATHGoogle Scholar
  20. 20.
    Shirosaki, M.: A hypersurface which determines linearly non-degenerrate holomorphic mappings. Kodai Math. 23, 105–107 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Yang, C.C., Hua, X.: Uniqueness polynomials of entire and meromorphic functions. Mat. Fiz. Anal. Geom. 3, 391–398 (1997)MathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Institute of MathematicsVASTHanoiVietnam
  2. 2.Thang Long UniversityHanoiVietnam
  3. 3.Hai Duong CollegeHai DuongVietnam
  4. 4.Thai Nguyen University of EducationThai NguyenVietnam

Personalised recommendations