On multiplicatively dependent vectors of polynomial values

Given polynomials f₁, , fₙ in m variables with integral coefficients, we give upper bounds for the number of integral m-tuples u₁, , uₙ of bounded height such that f₁ (u₁), , fₙ (uₙ) are multiplicatively dependent. We also prove, under certain conditions, a finiteness result for u Zᵐ with relatively prime entries such that f₁ (u), , fₙ (u) are multiplicatively dependent.