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**12 March 2014**

Let K be a function field of one variable over a constant field C of finite transcendence degree over ℂ. Let M/K be a finite extension and let W be a set of primes of K such that all but finitely many primes of W do not split in the extension M/K. Then there exists a set W′ of K-primes such that Hilbert's Tenth Problem is not decidable over OK,W′ = {x ϵ K∣ordþx ≥ 0, ∀þ ∉ W′}, and the set (W′ ∖ W)∪{W ∖ W′) is finite.

Let K be a function field of one variable over a constant field C finitely generated over ℚ. Let M/K be a finite extension and let W be a set of primes of K such that all but finitely many primes of W do not split in the extension M/K and the degree of all the primes in W is bounded by b ϵ ℕ. Then there exists a set W′ of K-primes such that ℤ has a Diophantine definition over OK,W′, and the set (W′ ∖ W)∪(W ∖ W′) is finite.

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