Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0 Laurent Moret-bailly and Alexandra Shlapentokh

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(x1, . . . , xn) has solutions in R.