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
x1,...,xn∈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.Comment: This version contains minor revisions and will appear in Annales de l
Institut Fourie