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

TL;DR

This paper provides effective bounds on $S$-integral preperiodic points for polynomials over number fields, advancing understanding of their finiteness and explicit bounds in specific cases like unicritical polynomials.

Contribution

It makes effective certain cases of Ih's conjecture on $S$-integral preperiodic points and derives explicit bounds for unicritical polynomials outside small parameter regions.

Findings

Bounds on the number of $S$-units in the sequence $^n(^m(eta)$

Explicit bounds depending on bad reduction places for unicritical polynomials

Novel lower bounds for $v$-adically smallest preperiodic points

Abstract

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 $K$.