Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

Details Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0 Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

2008-05-22

arxiv.org/abs/0805.3458v2

Mathematics

Logic

This version contains minor revisions and will appear in Annales de l Institut Fourier

Scientific paper

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 $\Q(x_1,...,x_n)$ has solutions in $R$.

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