### 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 $\mathbb{Q}({x}_{1},...,{x}_{n})$ has solutions in $R$.