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.
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Marley Young (2024) studied this question.
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