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Lower bounds on the canonical height associated to the morphism \(\phi(z)= z^d+c\). (English) Zbl 1239.11071
In this paper, the author proves a lower bound for the canonical height \({\hat h}_\phi(\alpha)\) of wandering algebraic points \(\alpha\) for rational functions \(\phi\) of a certain form. To describe it precisely, we will need some notation. Let \(K\) be a number field, and let \(\phi(x) = x^d + c\), where \(c \in K\) and \(d > 1\). We say that a finite place \(v\) is a place of Type II reduction for \(\phi\) if \(v(c) < 0\) and \(d | v(c)\). Then there is a constant \(C > 0\), depending only on \(K\), \(d\), and the number of Type II places of \(\phi\), such that \[ {\hat h}_\phi(\alpha) > C \max\{ h(c), 1\} \tag{1} \] for all \(\alpha \in K\) that are not preperiodic under \(\phi\).
The technique of the proof can be described as follows. First, one decomposes the canonical \({\hat h}_\phi\) into local heights \({\hat \lambda}_{\phi,v}\). If \(v\) is a finite place such that \(v(c) < 0\) and \(d\) does not divide \(v(c)\), then there is a simple explicit lower bound for \(\lambda_{\phi,v}(\alpha)\) in terms of \(\log |c|_v\). Hence, one need only treat the archimedean places and the places of Type II reduction. At the archimedean and Type II places, one can show that if two points are close together, then at least one of them must have large height, via the product formula (intuitively, two rational numbers can only be close together if one of them has a large denominator). If many iterates of \(\alpha\) have reasonably small \(v\)-adic absolute value, then some must be \(v\)-adically near each other, by the pigeon hole principle, and one thus obtains a lower bound on \({\hat h}_\phi(\alpha)\).
The lower bound [J. H. Silverman, Duke Math. J. 48, 633–648 (1981; Zbl 0475.14033)] is analogous to a lower bound proved by Silverman [loc. cit.] for non-torsion points of elliptic curves. There, the bound depends on the number of places of multiplicative reduction. Lang has conjectured [M. Hindry and J. H. Silverman, Invent. Math. 93, No. 2, 419–450 (1988; Zbl 0657.14018)] that such a bound exists for non-torsion points of elliptic curve without any dependence on the number of places of multiplicative reduction; Hindry and Silverman [loc. cit.] have proved that this conjecture does hold over function fields. In this paper, the author suggests a dynamical analog of Lang’s conjecture. Namely he asks if, given a number field \(K\), and a polynomial \(\phi(x) = x^d + c\) with \(c \in K\) and \(d > 1\), there is a constant \(C > 0\), depending only on \(d\) and \(K\) (and not on any notion of places of bad reduction), such that \[ {\hat h}_\phi(\alpha) > C \max\{ h(\alpha), 1 \} \] for all \(\alpha \in K\) that are not preperiodic for \(\phi\).

MSC:
11G50 Heights
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
11G99 Arithmetic algebraic geometry (Diophantine geometry)
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