A lower bound on the canonical height for polynomials

  • Nicole Looper


We prove a lower bound on the canonical height associated to polynomials over number fields evaluated at points with infinite forward orbit. The lower bound depends only on the degree of the polynomial, the degree of the number field, and the number of places of bad reduction.

Mathematics Subject Classification

11G50 37P30 37P45 37P15 



I would like to thank Laura DeMarco for introducing me to the trees in [6] and [7], and for many helpful and enlightening discussions on that topic. I would also like to thank Patrick Ingram for several fruitful conversations on this research problem, and for sharing his results concerning the normal form used throughout this article. Finally, it is a pleasure to thank Laura DeMarco and Joseph Silverman for their useful comments on an earlier draft, and the anonymous referee for his or her careful reading and additional suggestions.


  1. 1.
    Baker, M., Payne, S., Rabinoff, J.: On the structure of non-Archimedean analytic curves. Contemp. Math. 605, 93–121 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Benedetto, R., Dickman, B., Joseph, S., Krause, B., Rubin, D., Zhou, X.: Computing points of small height for cubic polynomials. Involve 2, 37–64 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Benedetto, R.: Preperiodic points of polynomials over global fields. Journal für die Reine und Angewandte Mathematik 608, 123–153 (2007)MathSciNetMATHGoogle Scholar
  4. 4.
    Branner, B., Hubbard, J.: The iteration of cubic polynomials. II. Patterns and parapatterns. Acta Math. 169, 229–325 (1992)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Call, G., Silverman, J.: Canonical heights on varieties with morphisms. Compos. Math. 89(2), 163–205 (1993)MathSciNetMATHGoogle Scholar
  6. 6.
    DeMarco, L.: Finiteness for degenerate polynomials. In: Lyubich, M., Yampolsky, M. (eds.) Holomorphic Dynamics and Renormalization: A Volume in Honour of John Milnor’s 75th Birthday, vol 53, pp. 89–104. Fields Institute Communications, AMS, Toronto (2008)Google Scholar
  7. 7.
    DeMarco, L., McMullen, C.: Trees and the dynamics of polynomials. Annales Scientifiques de l’École Normale Supérieure 41, 337–383 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hindry, M., Silverman, J.: The canonical height and integral points on elliptic curves. Inventiones Mathematicae 93, 419–450 (1998)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ingram, P.: A finiteness result for post-critically finite polynomials. Int. Math. Res. Not. 3, 524–543 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ingram, P.: Lower bounds on the canonical height associated to the morphism \(z^d+c\). Monatschefte für Mathematik 157, 69–89 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lang, S.: Elliptic Curves: Diophantine Analysis. Grundlehren der Mathematischen Wissenschaften, vol. 231. Springer, Berlin (1978)MATHGoogle Scholar
  12. 12.
    McMullen, C.: Complex Dynamics and Renormalization. Annals of Mathematical Studies, vol. 135. Princeton University Press, Princeton (1994)MATHGoogle Scholar
  13. 13.
    Milnor, J.: Dynamics in One Complex Variable. Annals of Mathematical Studies, vol. 10. Princeton University Press, Princeton (2006)MATHGoogle Scholar
  14. 14.
    Remmert, R.: Classical Topics in Complex Function Theory. Graduate Texts in Mathematics, vol. 172. Springer, New York (1998)MATHGoogle Scholar
  15. 15.
    Silverman, J.: The Arithmetic of Dynamical Systems. Graduate Texts in Mathematics, vol. 241. Springer, New York (2007)MATHGoogle Scholar
  16. 16.
    Silverman, J.: Lower bound for the canonical height on elliptic curves. Duke Math. J. 48, 633–648 (1981)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Silverman, J.: Moduli Spaces and Arithmetic Dynamics. CRM Monograph Series, vol. 30. AMS, Providence (2012)MATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA

Personalised recommendations