Dynamical uniform boundedness and the \(abc\)-conjecture (Q2041855)

The conjecture of \textit{P. Morton} and \textit{J. H. Silverman} [Int. Math. Res. Not. 1994, No. 2, 97--109 (1994; Zbl 0819.11045)] implies that if \(K\) is an algebraic number field of degree \(D\) and \(f\in K[x]\) is of degree \(d\ge 2\), then \(f\) has at most \(B\) preperiodic points in \(K\), where \(B\) depends only on \(D\) and \(d\). In its weaker form the bound \(B\) is allowed to depend on \(d\) and the field \(K\). This conjecture is still wide open. The author shows that for polynomials of the form \(x^d+c\) the weak form of this conjecture is a consequence of the \(ABC\)-conjecture for algebraic number fields. The presented proof covers also the case when \(K\) is a one-dimensional function field of characteristic zero. The last assertion is now known to hold unconditionally (see [\textit{J. R. Doyle} and \textit{B. Poonen}, Compos. Math. 156, No. 4, 733--743 (2020; Zbl 1445.37080), Theorem 1.7]). In the case \(d\ge5\) the proof uses the standard \(ABC\)-conjecture for number fields, whereas for \(d=2,3,4\) a generalized form of this conjecture is applied, shown by the author in Sect. 2.2 to be a consequence of Conjecture 2.3 by \textit{P. Vojta} [Int. Math. Res. Not. 1998, No. 21, 1103--1116 (1998; Zbl 0923.11059)]. Recently \textit{C. Panraksa} [``Rational periodic points of $x^d+c$ and Fermat-Catalan equations''; Preprint, \url{arXiv:2105.03715}] gave a simple proof of this theorem in the case when \(K\) is the field of rational numbers and \(d\) is sufficiently large. If \(d=p^m\) is an odd prime power and \(K\) is a field not containing the \(p\)-th roots of unity, then the weak form of the Morton-Silverman conjecture holds unconditionally for polynomials \(x^d+c\in K[x]\) [the reviewer, Funct. Approx. 42, 163--168 (2020; Zbl.1262.11090)].