Canonical heights and preperiodic points for weighted homogeneous families of polynomials

TL;DR

This paper establishes lower bounds on canonical heights for weighted homogeneous polynomial families over number fields, linking height to bad reduction and providing uniform bounds on preperiodic points, with some results conditional on the abc conjecture.

Contribution

It introduces a lower bound on canonical heights for weighted homogeneous polynomial families and proves uniform bounds on preperiodic points, extending understanding of dynamical systems over number fields.

Findings

Lower bounds on canonical height depending on bad reduction

Uniform bounds on preperiodic points unconditionally

Conditional absolute bounds assuming the abc conjecture

Abstract

A family $f_t(z)$ of polynomials over a number field $K$ will be called \emph{weighted homogeneous} if and only if $f_t(z)=F(z^e, t)$ for some binary homogeneous form $F(X, Y)$ and some integer $e\geq 2$. For example, the family $z^d+t$ is weighted homogeneous. We prove a lower bound on the canonical height, of the form \[\hat{h}_{f_t}(z)\geq \epsilon \max\{h_{\mathsf{M}_d}(f_t), \log|\operatorname{Norm}\mathfrak{R}_{f_t}|\},\] for values $z\in K$ which are not preperiodic for $f_t$. Here $\epsilon$ depends only on the number of places at which $f_t$ 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 $\varphi$ over $K$ (assuming the $abc$ Conjecture), as well as uniform bounds on the number of preperiodic points (unconditionally).