Let $\Gamma$ be a finitely generated subgroup of $C^*$. For a polynomial $P\in Z[x_1,...x_k]$, determine whether $P(x_1,...x_k)=0$ has a zero in $\Gamma$. Is this decidable?

Motivation is related to the question whether whether there are standard examples of stable theories that are undecidable: the theory of $(C,+,*, \Gamma)$ (that is, we add a predicate for the subgroup $\Gamma$) is superstable, and a positive answer is necessary (and perhaps sufficient) for this theory to be decidable. With a similar motivation (the superstable theory of generic powers where you add to an algebraically closed field multivalued functions $x^\alpha$ where $\alpha$ is sufficiently generic), one can ask similar but complicated number theoretic questions related to Faltings' and Ax's theorems. $\Gamma$.