Abstract
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 ∈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.
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