  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.
