JournalsjemsVol. 24, No. 5pp. 1567–1592

Rational values of transcendental functions and arithmetic dynamics

  • Gareth Boxall

    Stellenbosch University, South Africa
  • Gareth A. Jones

    University of Manchester, UK
  • Harry Schmidt

    University Basel, Switzerland
Rational values of transcendental functions and arithmetic dynamics cover
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Abstract

We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work with pp-adic methods to obtain, for each positive ε\varepsilon, an upper bound of the form cD3n/4+εncD^{3n/4 + \varepsilon n} on the number of irreducible factors of Pn(X)Pn(α)P^{\circ n}(X)-P^{\circ n}(\alpha) over KK, where KK is a number field, PP is a polynomial of degree D2D\geq 2 over KK, PnP^{\circ n} is the nn-th iterate of PP, α\alpha is a point in KK for which {Pn(α):nN}\{P^{\circ n}(\alpha):n\in\mathbb{N}\} is infinite and cc depends effectively on P,α,[K:Q]P, \alpha, [K:\mathbb{Q}] and ε\varepsilon.

Cite this article

Gareth Boxall, Gareth A. Jones, Harry Schmidt, Rational values of transcendental functions and arithmetic dynamics. J. Eur. Math. Soc. 24 (2022), no. 5, pp. 1567–1592

DOI 10.4171/JEMS/1120