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Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

Laurent Moret-Bailly;  Alexandra Shlapentokh — 2009

Annales de l’institut Fourier

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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