Transcendence of polynomial canonical heights

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.