A Diophantine problem for Laurent polynomial rings

Let $R$ be an integral domain of characteristic zero. We prove that the diophantine problem for the Laurent polynomial ring $R[T,{T^{ - 1}}]$ with coefficients in ${\mathbf{Z}}[T]$ is unsolvable. Under suitable conditions on $R$ we then show that either ${\mathbf{Z}}$ or ${\mathbf{Z}}[i]$ is diophantine over $R[T,{T^{ - 1}}]$.