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 -adic methods to obtain, for each positive , an upper bound of the form on the number of irreducible factors of over , where is a number field, is a polynomial of degree over , is the -th iterate of , is a point in for which is infinite and depends effectively on and .
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. 1567–1592DOI 10.4171/JEMS/1120