On multiplicative dependence between elements of polynomial orbits

We classify the pairs of polynomials f, g CX 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 f₁^m₁ (0), , fₙ^mₙ (0) are multiplictively dependent are sparse. We also classify the pairs f, g QX such that there are infinitely many (x, y) Z² satisfying f (x) ᵏ=g (y) ^ for some (possibly varying) non-zero integers k,.