Ramification in iterated towers for rational functions (Q663315)

Let \(K\) be a number field and let \(\phi(x)\) be a rational function of degree greater than 1 defined over \(K\). Let \(\Phi_n(x,t)=\phi^{(n)}(x)-t\) where \(\phi^{(n)}\) is the \(n\)th iterate of \(\phi\). The authors give a formula for the discriminant of the numerator of \(\Phi_n(x,t)\). If \(\phi\) is \textit{postcritically finite}, i.e. if the forward orbit of the critical points of \(\phi\) under all iterations is a finite set, then the authors also prove that for each specialization of \(t\) to \(t_0 \in K\), then there exists a finite set of primes of \(K\) containing the prime divisors of the discriminant of \(\Phi_n\) for all \(n\).