Lower Bounds for the Canonical Height of a Unicritical Polynomial and Capacity

TL;DR

This paper extends Dimitrov's approach to establish lower bounds for the canonical height of points in polynomial dynamical systems of the form T^p + c, especially when the orbit of zero is finite, revealing inverse relationships with field degree.

Contribution

It adapts Dimitrov's method to polynomial dynamical systems with finite zero orbit, providing explicit lower bounds for canonical heights in these systems.

Findings

Lower bounds for local canonical height inversely proportional to field degree.

Lower bounds for canonical height decay quadratically with field degree.

Application to specific polynomial systems like T^2 + c with finite zero orbit.

Abstract

In a recent breakthrough, Dimitrov solved the Schinzel-Zassenhaus Conjecture. We follow his approach and adapt it to certain dynamical systems arising from polynomials of the form $T^p+c$ where $p$ is a prime number and where the orbit of $0$ is finite. For example, if $p=2$, and $0$ is periodic under $T^2+c$ with $c\in\mathbb{R}\smallsetminus\{-2\}$, we prove a lower bound for the local canonical height of a wandering algebraic integer that is inversely proportional to the field degree. From this we are able to deduce a lower bound for the canonical height of a wandering point that decays like the inverse square of the field degree.