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