Let $d\geq 2$ and let $H$ denote the absolute multiplicative Weil height on $\bar{\mathbb{Q}}$. Let $f(z)\in \bar{\mathbb{Q}}[z]$ of degree $d$ and let $a\in\bar{\mathbb{Q}}$, the multiplicative and logarithmic canonical heights of $a$ with respect to $f$ are defined as
$$ \hat{H}_f (a) =\lim H(f^n(a))^{1/d^n} \quad \text{and}\quad \hat{h}_f(a)=\log \hat{H}_f(a). $$
Let $n$ be a positive integer. For $1\leq i\leq n$, let $f_i\in\bar{\mathbb{Q}}[z]$ of degree $d$ and let $a_i\in\bar{\mathbb{Q}}$. In this talk, we provide a complete characterization of when the $\hat{H}_{f_i}(a_i)$’s are multiplicatively dependent modulo constant meaning there exist integers $m_1,\ldots,m_n$ not all of which are $0$ and $a\in \bar{\mathbb{Q}}$ such that:
$$ \hat{H}_{f_1}(a_1)^{m_1} \cdots \hat{H}_{f_n}(a_n)^{m_n}=a. $$
As an immediate consequence, we characterize all the pairs $(f,a)$ such that $\hat{H}_f(a)$ is an algebraic number and proves the existence of $(f,a)$ such that $\hat{h}_f(a)$ is an irrational number. The proof uses the Medvedev-Scanlon classification of preperiodic subvarieties under the dynamics of a split polynomial map and the construction of a certain auxiliary polynomial. This is joint work with Jason Bell.