On diophantine definability and decidability in some rings of algebraic functions of characteristic

Journal of Symbolic Logic 67 (2):759-786 (2002)
Abstract
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 $O_{K,W'} = \{x \in K\mid ord_\mathfrak{p} x \geq 0, \forall\mathfrak{p} \notin W'\}$ , and the set (W' $\backslash$ W) ∪ (W $\backslash$ 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 ∈ N. Then there exists a set W' of K-primes such that Z has a Diophantine definition over O K ,W', and the set (W' $\backslash$ W) ∪ (W $\backslash$ W') is finite
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