Counting multiplicatively dependent vectors of algebraic numbers

Given a vector $v=(v_1, \ldots, v_n)$ of $n$ complex numbers, we say that $v$ is multiplicatively dependent if there is a non-zero integer vector $k=(k_1, \ldots, k_n)$ such that $v_1^{k_1} \cdots v_n^{k_n}=1$. In this talk, I will present some recent results on counting multiplicatively dependent vectors of algebraic numbers of fixed degree (or within a number field) and bounded height. These include sharp lower and upper bounds, and especially asymptotic formulas for several cases. (This is joint work with Francesco Pappalardi, Igor Shparlinski and Cameron Stewart.)