The $abcd$ conjecture, uniform boundedness, and dynamical systems

Abstract

We survey Vojta's higher-dimensional generalizations of the abc conjecture and Szpiro's conjecture as well as recent developments that apply them to various problems in arithmetic dynamics. In particular, the "abcd conjecture" implies a dynamical analogue of a conjecture on the uniform boundedness of torsion points and a dynamical analogue of Lang's conjecture on lower bounds for canonical heights.

A Dynamical System Proof of Niven’s Theorem and Its Extensions

“…For polynomials with integral coefficients, by using a simple divisibility argument, it can be shown that all preperiodic points have periods at most 2. For a number field K and c ∈ O K , a result in [4] implies that the set of preperiodic points of f c (x) is uniformly bounded depending only on D = [K : Q] (see also [22]). However, it is still interesting to find an explicit bound for the number of preperiodic points of an integral quadratic polynomial.…”