Advertisement

The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1032–1042 | Cite as

Upper Bounds of GCD Counting Function for Holomorphic Maps

  • Xiaojun Liu
  • Guangsheng YuEmail author
Article
  • 90 Downloads

Abstract

In this paper, we give upper bounds for the gcd counting function(which is an analogue for the notion of gcd in the context of holomorphic maps) in various settings. As applications, we obtain analytic dependence of entire functions from the second main theorem and multiplicative dependence under the fundamental conjecture for entire curves.

Keywords

Gcd counting function Analytic dependence Multiplicative dependence Second main theorem 

Mathematics Subject Classification

11A05 30D35 32A22 32H30 

Notes

Acknowledgements

X. Liu was supported by NSFC11401381, G. Yu was supported by NSFC11671090.

References

  1. 1.
    Brown, A.B.: Functional dependence. Trans. Am. Math. Soc. 38, 379–394 (1935)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Corvaja, P., Zannier, U.: A lower bound for the height of a rational function at S-unit points. Monatsh. Math. 144(3), 203–224 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Griffiths, P., King, J.: Nevanlinna theory and holomorphic mappings between algebraic varieties. Acta Math. 130, 145–220 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ji, Q.C., Yu, G.S.: On the second main theorem of Nevanlinna theory for singular divisors with \((k,\ell )\)-conditions. Asian J. Math. (to appear)Google Scholar
  5. 5.
    Lang, S.: Fundamentals of Diophantine Geometry. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  6. 6.
    Noguchi, J., Winkelmann, J.: Nevanlinna Theory in Several Complex Variables and Diophantine Approximation. Springer, Berlin (2013)zbMATHGoogle Scholar
  7. 7.
    Noguchi, J., Winkelmann, J., Yamanoi, K.: Degeneracy of holomorphic curves into algebraic varieties. J. Math. Pures Appl. (9) 88(3), 293–306 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Noguchi, J., Winkelmann, J., Yamanoi, K.: The second main theorem for holomorphic curves into semi-abelian varieties. II. Forum Math. 20(3), 469–503 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Pasten, H., Wang, J.T.-Y.: GCD Bounds for analytic functions. Int. Math. Res. Not. 2017(1), 47–95 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Shiffman, B.: Nevanlinna defect relations for singular divisors. Invent. Math. 31(2), 155–182 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Silverman, J.H.: Generalized greatest common divisors, divisibility sequences, and Vojta’s conjecture for blowups. Monatsh. Math. 145(4), 333–350 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Vojta, P.: Diophantine Approximations and Value Distribution Theory. Lecture Notes in Mathematics, vol. 1239. Springer, Berlin (1987)zbMATHGoogle Scholar
  13. 13.
    Yamanoi, K.: Holomorphic curves in abelian varieties and intersections with higher codimensional subvarieties. Forum Math. 16(5), 749–788 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Zannier, U.: Diophantine equations with linear recurrences. An overview of some recent progress. J. Théor. Nombres Bordeaux 17(1), 423–435 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Shanghai for Science and TechnologyShanghaiPeople’s Republic of China

Personalised recommendations