The Diophantine equation $\alpha _{1}^{x_{1}}\cdots \alpha _{n}^{x_{n}} = f(x_{1},\dots ,x_{n})$. II

We will deal with the equation of the title where $\alpha _{1},\dots ,\alpha _{n}$ are multiplicatively independent complex numbers and $f$ is a polynomial. We will give a bound for the number of solutions which depends only on $n$ and the degree of $f$. Two further results which play a rôle in the proof are of independent interest.