On multiplicatively dependent vectors of polynomial values

Given polynomials $f_1,\dots ,f_n$ in $m$ variables with integral coefficients, we give upper bounds for the number of integral $m$-tuples ${\mathbf{u}}_1,\dots ,{\mathbf{u}}_n$ of bounded height such that $f_1({\mathbf{u}}_1),\dots ,f_n({\mathbf{u}}_n)$ are multiplicatively dependent. We also prove, under certain conditions, a finiteness result for $\mathbf{u}\in {\mathbb{Z}}^m$ with relatively prime entries such that $f_1(\mathbf{u}),\dots ,f_n(\mathbf{u})$ are multiplicatively dependent.