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Lower bounds on the canonical height associated to the morphism \({\phi(z)= z^d+c}\)

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Abstract

A lower bound is obtained of the canonical height associated to the morphism \({\phi(z)=z^d+c}\) evaluated at wandering points α. The lower bound is of the form Ch(c), for some constant C depending on the the number of primes of bad reduction for \({\phi}\) , and the degree of the number field \({\mathbb{Q}(c, \alpha)}\) .

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References

  1. Baker M.: A lower bound for average values of dynamical Green’s functions. Math. Res. Lett. 13, 245–257 (2006)

    MATH  MathSciNet  Google Scholar 

  2. Benedetto, R.: Pre-periodic points of polynomials over global fields, arXiv:math/0506480

  3. Call G.S., Silverman J.H.: Canonical heights on varieties with morphisms. Composit. Math. 89, 163–205 (1993)

    MATH  MathSciNet  Google Scholar 

  4. Hindry M., Silverman J.H.: The canonical height and integral points on elliptic curves. Invent. Math. 93, 419–450 (1998)

    Article  MathSciNet  Google Scholar 

  5. Poonen B.: The classification of rational pre-periodic points of quadratic polynomials over \({\mathbb{Q}}\) : a refined conjecture. Math. Z. 228, 11–29 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Silverman J.H.: The Arithmetic of Elliptic Curves. Springer, Heidelberg (1985)

    Google Scholar 

  8. Silverman J.H.: The Arithmetic of Dynamical Systems. Springer, Heidelberg (2007)

    MATH  Google Scholar 

  9. Szpiro, L., Tucker, T.: Equidistribution and generalized Mahler measures. arXiv:math/0510404

  10. Szpiro, L., Tucker, T.: A Shafarevich–Faltings theorem for rational functions. arXiv:math/0603436

  11. Walde R., Russo P.: Rational periodic points of the quadratic function Q c  = x 2 + c. Am. Math. Mon. 101, 318–331 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, Canada

    Patrick Ingram

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  1. Patrick Ingram
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Correspondence to Patrick Ingram.

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Communicated by U. Zannier.

The author was supported by a PDF grant from NSERC.

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Ingram, P. Lower bounds on the canonical height associated to the morphism \({\phi(z)= z^d+c}\) . Monatsh Math 157, 69–89 (2009). https://doi.org/10.1007/s00605-008-0018-6

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  • DOI: https://doi.org/10.1007/s00605-008-0018-6

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