Lower bounds on the canonical height associated to the morphism φ ( z ) = z d + c

where h is the usual absolute logarithmic height. It is reasonably easy to show, from the two properties above, that ĥφ(α) = 0 just in case φ j(α) = φi(α) for some j 6= i. If this is the case, we will say that α is a pre-periodic point for φ. It is natural to ask how small the value of ĥφ can be at points which are not pre-periodic, i.e., wandering points for φ. We will examine this question for morphisms for the form φ(z) = zd + c, for d ≥ 2. The canonical heights mentioned above are analogous to the canonical heights on elliptic curves (and more general abelian varieties) studied by Néron and Tate. The analogous question in this context, namely how small the canonical height of a non-torsion point on an elliptic curve may be, is the subject of a conjecture of Lang. Specifically, Lang conjectured that the height of such a point is bounded below by a constant multiple of max{h(jE), log |NormK/Q DE/K |, 1}, where jE and DE/K are the j-invariant and minimal discriminant of E/K respectively (see [5] for definitions of these terms). Silverman [4] has given a partial solution to this conjecture, proving that (for a non-torsion point P on an elliptic curve E)