On multiplicative dependence between elements of polynomial orbits

Abstract

We classify the pairs of polynomials $f,g \in \mathbb{C}[X]$ having orbits satisfying infinitely many multiplicative dependence relations, extending a result of Ghioca, Tucker and Zieve. Moreover, we show that given $f_1,\ldots, f_n$ from a certain class of polynomials with integer coefficients, the vectors of indices $(m_1,\ldots,m_n)$ such that $f_1^{m_1}(0),\ldots,f_n^{m_n}(0)$ are multiplictively dependent are sparse. We also classify the pairs $f,g \in \mathbb{Q}[X]$ such that there are infinitely many $(x,y) \in \mathbb{Z}^2$ satisfying $f(x)^k=g(y)^\ell$ for some (possibly varying) non-zero integers $k,\ell$.