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On Dipphantine definability and decidability in some rings of algebraic functions of characteristic 0

Published online by Cambridge University Press:  12 March 2014

Alexandra Shlapentokh
Alexandra Shlapentokh*
Affiliation:
East Carolina University, Department of Mathematics, Greenville, NC 27858, USA, E-mail: shlapentokha@mail.ecu.edu
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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 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 2 , June 2002 , pp. 759 - 786
Copyright
Copyright © Association for Symbolic Logic 2002

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