Effective integrality results in arithmetic dynamics
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Given a rational function f defined over a number field K, S. Ih conjectured the finiteness of f-preperiodic points which are S-integral relative to a given non-preperiodic point β. This conjecture remains open, but certain special cases have been proved. We formulate a generalisation of Ih's conjecture, considering a semigroup
Given an arbitrary, finitely generated rational semigroup
In the case of a single, unicritical polynomial f_c(z)=z^d+c, with β set to be the critical point 0, for parameters c outside a small region, we give an explicit bound which depends only on the number of places of bad reduction for f_c. As part of the proof, we obtain novel lower bounds for the v-adically smallest preperiodic point of f_c for each place v of K.
Finally, when