Canonical Heights and Preperiodic Points for Certain Weighted Homogeneous Families of Polynomials

Abstract

Abstract A family f of polynomials over a number field K will be called weighted homogeneous if and only if ft(z) = F(ze, t) for some binary homogeneous form F(X, Y) and some integer e ≥ 2. For example, the family zd + t is weighted homogeneous. We prove a lower bound on the canonical height, of the form \begin{align*} \hat{h}_{f_{t}}(z)\geq \varepsilon \max\!\left\{h_{\mathsf{M}_{d}}(f_{t}), \log|\operatorname{Norm}\mathfrak{R}_{f_{t}}|\right\},\end{align*} for values z ∈ K which are not preperiodic for ft. Here ε depends only on the number field K, the family f, and the number of places at which ft has bad reduction. For suitably generic morphisms $\varphi :\mathbb {P}^{1}\to \mathbb {P}^{1}$, we also prove an absolute bound of this form for t in the image of φ over K (assuming the abc Conjecture), as well as uniform bounds on the number of preperiodic points (unconditionally).