Abstract
Let \({\phi}\) be a rational function of degree at least two defined over a number field k. Let \({a \in \mathbb{P}^1(k)}\) and let K be a number field containing k. We study the cardinality of the set of rational iterated preimages Preim\({(\phi, a, K) = \{x_{0} \in \mathbb{P}^1(K) | \phi^{N} (x_0) = a {\rm for some} N \geq 1\}}\). We prove two new results (Theorems 2 and 4) bounding \({|{\rm Preim}(\phi, a, K)|}\) as \({\phi}\) varies in certain families of rational functions. Our proofs are based on unit equations and a method of Runge for effectively determining integral points on certain affine curves. We also formulate and state a uniform boundedness conjecture for Preim\({(\phi, a, K)}\) and prove that a version of this conjecture is implied by other well-known conjectures in arithmetic dynamics.
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Communicated by U. Zannier.
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Levin, A. Rational preimages in families of dynamical systems. Monatsh Math 168, 473–501 (2012). https://doi.org/10.1007/s00605-012-0426-5
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DOI: https://doi.org/10.1007/s00605-012-0426-5