On multiplicative dependence between elements of polynomial orbits

We classify the pairs of polynomials f,g ∈ C[X] having orbits satisfying infinitely many multiplicative dependence relations, extending a result of Ghioca, Tucker and Zieve. Moreover, we show that given f₁,…, fₙ from a certain class of polynomials with integer coefficients, the vectors of indices (m₁,…,mₙ) such that f₁m₁(0),…,fₙmₙ(0) are multiplictively dependent are sparse. We also classify the pairs f,g ∈ Q[X] such that there are infinitely many (x,y) ∈ Z² satisfying f(x)ᵏ=g(y)^ for some (possibly varying) non-zero integers k,.