Transcendence of polynomial canonical heights

TL;DR

This paper investigates the algebraic nature of canonical heights in polynomial dynamics, solving the problem of characterizing when these heights are algebraic and making progress on the conjecture about their transcendence or rationality.

Contribution

It characterizes all pairs (f,a) with algebraic canonical heights and advances understanding of the transcendence of heights in polynomial dynamical systems.

Findings

Solved the problem of characterizing algebraic canonical heights for polynomial maps.

Established conditions under which canonical heights are algebraic.

Used Medvedev-Scanlon classification to analyze preperiodic subvarieties.

Abstract

There are two fundamental problems motivated by Silverman's conversations over the years concerning the nature of the exact values of canonical heights of $f(z)\in\bar{\mathbb{Q}}(z)$ where $f$ has degree $d\geq 2$. The first problem is the conjecture that $\hat{h}_f(a)$ is either $0$ or transcendental for every $a\in \mathbb{P}^1(\bar{\mathbb{Q}})$; this holds when $f$ is linearly conjugate to $z^d$ or $\pm C_d(z)$ where $C_d(z)$ is the Chebyshev polynomial of degree $d$ since $\hat{H}_f(a)$ is algebraic for every $a$. Other than this, very little is known: for example, it is not known if there \emph{exists} even \emph{one} rational number $a$ such that $\hat{h}_f(a)$ is \emph{irrational} where $f(z)=z^2+\displaystyle\frac{1}{2}$. The second problem asks for the characterization of all pairs $(f,a)$ such that $\hat{H}_f(a)$ is algebraic. In this paper, we solve the second problem and obtain significant progress to the first problem in the case of polynomial dynamics. These are consequences of our main result concerning the possible algebraic numbers that can be expressed as a multiplicative combination of values of B\"ottcher coordinates. The proof of our main result uses a construction of a certain auxiliary polynomial and the powerful Medvedev-Scanlon classification of preperiodic subvarieties of split polynomial maps.