Skip to main content
SpringerLink
Log in

Dynamical uniform boundedness and the abc-conjecture

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We address the Uniform Boundedness Conjecture of Morton and Silverman in the case of unicritical polynomials, assuming a generalization of the abc-conjecture. For unicritical polynomials of degree at least five, we require only the standard abc-conjecture.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Baker, M., Payne, S., Rabinoff, J.: On the structure of nonarchimedean analytic curves. Contemp. Math. 605, 93–121 (2013)

    Article  Google Scholar 

  2. Baker, M., Rumely, R.: Potential Theory and Dynamics on the Berkovich Projective Line. Mathematical Surveys and Monographs, vol. 159. AMS, Providence (2010)

    Book  Google Scholar 

  3. Benedetto, R.: Preperiodic points of polynomials over global fields. J. Reine Angew. Math. 608, 123–153 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Brownawell, W.D., Masser, D.W.: Vanishing sums in function fields. Math. Proc. Camb. Philos. Soc. 100(3), 427–434 (1986)

    Article  MathSciNet  Google Scholar 

  5. DeMarco, L., Faber, X.: Degenerations of complex dynamical systems. Forum Math. Sigma 2, e6 (2014)

    Article  MathSciNet  Google Scholar 

  6. DeMarco, L., Faber, X.: Degenerations of complex dynamical systems II: analytic and algebraic stability. Math. Ann. 365, 1669–1699 (2016)

    Article  MathSciNet  Google Scholar 

  7. DeMarco, L., Rumely, R.: Transfinite diameter and the resultant. J. Reine Angew. Math. 611, 145–161 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Doyle, J., Poonen, B.: Gonality of dynatomic curves and strong uniform boundedness of preperiodic points. Compos. Math. 156, 733–743 (2020)

    Article  MathSciNet  Google Scholar 

  9. Frey, G.: Elliptic Curves and Solutions of \(A-B=C\). Séminaire de Theéorie des Nombres, Paris 1985–86, Volume 71 of, pp. 39–51. Birkhaüser Boston, Boston (1987)

    Book  Google Scholar 

  10. Frey, G.: Links Between Solutions of \(A-B=C\) and Elliptic Curves. Number Theory, Volume 1380 of Lecture Notes in Mathematics, pp. 31–62. Springer, New York (1989)

    Google Scholar 

  11. Hindry, M., Silverman, J.H.: The canonical height and integral points on elliptic curves. Invent. Math. 93, 419–450 (1988)

    Article  MathSciNet  Google Scholar 

  12. Hindry, M., Silverman, J.H.: Diophantine Geometry: An Introduction. Graduate Texts in Mathematics, vol. 201. Springer, Berlin (2000)

    Book  Google Scholar 

  13. Ingram, P.: Lower bounds on the canonical height associated to the morphism \(f(z)=z^d+c\). Monatschefte Math. 157, 69–89 (2007)

    Article  Google Scholar 

  14. Looper, N.R.: A lower bound on the canonical height for polynomials. Math. Ann. 373, 1057–1074 (2019)

    Article  MathSciNet  Google Scholar 

  15. Mazur, B.: Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math. 47, 33–186 (1997)

    Article  MathSciNet  Google Scholar 

  16. Merel, L.: Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124(1–3), 437–449 (1996)

    Article  MathSciNet  Google Scholar 

  17. Morton, P., Silverman, J.H.: Periodic points, multiplicities, and dynamical units. J. Reine Angew. Math. 461, 81–122 (1995)

    MathSciNet  MATH  Google Scholar 

  18. Morton, P., Silverman, J.H.: Rational periodic points of rational functions. Int. Math. Res. Not. 2, 97–110 (1994)

    Article  MathSciNet  Google Scholar 

  19. Narkiewicz, W.: Polynomial cycles in algebraic number fields. Colloq. Math. 58(1), 151–155 (1989)

    Article  MathSciNet  Google Scholar 

  20. Silverman, J.H.: The Arithmetic of Dynamical Systems. Graduate Texts in Mathematics, vol. 241. Springer, New York (2007)

    Book  Google Scholar 

  21. Silverman, J.H.: Lower bound for the canonical height on elliptic curves. Duke Math. J. 48, 633–648 (1981)

    Article  MathSciNet  Google Scholar 

  22. Vojta, P.: A more general \(abc\) conjecture. Int. Math. Res. Not. 1998, 1103–1116 (1998)

    Article  MathSciNet  Google Scholar 

  23. Voloch, J.F.: Diagonal equations over function fields. Bol. Soc. Brasil. Mat. 16(2), 29–39 (1985)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I would like to thank Rob Benedetto, Laura DeMarco, Holly Krieger, Joe Silverman, and Tom Tucker for useful discussions relating to this project. I thank Holly Krieger in particular for extensive and fruitful conversations regarding the arguments presented in Sect. 3, and Joe Silverman for his many helpful comments on a draft of this article. I would also like to thank the anonymous referees for their useful suggestions improving the clarity of the exposition.

Author information

Authors and Affiliations

  1. Brown University, Providence, RI, USA

    Nicole R. Looper

Authors
  1. Nicole R. Looper
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicole R. Looper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author’s research was supported by NSF Grant DMS-1803021.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Looper, N.R. Dynamical uniform boundedness and the abc-conjecture. Invent. math. 225, 1–44 (2021). https://doi.org/10.1007/s00222-020-01029-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-020-01029-7

Search

Navigation