Effective bounds on $S$-integral preperiodic points for polynomials

Given a polynomial $f$ defined over a number field $K$, we make effective certain special cases of a conjecture of S. Ih, on the finiteness of $f$-preperiodic points which are $S$-integral with respect to a fixed non-preperiodic point $\alpha$. As an application, we obtain bounds on the number of $S$-units in the doubly indexed sequence $\{ f^n(\alpha) - f^m(\alpha) \}_{n > m \geq 0}$. In the case of a unicritical polynomial $f_c(z)=z^2+c$, with $\alpha$ fixed 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 $K$.