Diophantine Undecidability of Holomorphy Rings of Function Fields of   Characteristic 0
  Let $K$ be a one-variable function field over a field of constants of characteristic 0. Let $R$ be a holomorphy subring of $K$, not equal to $K$. We prove the following undecidability results for $R$: If $K$ is recursive, then Hilbert's Tenth Problem is undecidable in $R$. In general, there exist $x_1,...,x_n \in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in $\Q(x_1,...,x_n)$ has solutions in $R$. 
Laurent Moret-Bailly, Alexandra Shlapentokh