Diophantine equivalence and countable rings

Journal of Symbolic Logic 59 (3):1068-1095 (1994)
We show that Diophantine equivalence of two suitably presented countable rings implies that the existential polynomial languages of the two rings have the same "expressive power" and that their Diophantine sets are in some sense the same. We also show that a Diophantine class of countable rings is contained completely within a relative enumeration class and demonstrate that one consequence of this fact is the existence of infinitely many Diophantine classes containing holomophy rings of Q
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DOI 10.2307/2275929
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Alexandra Shlapentokh (1996). Rational Separability Over a Global Field. Annals of Pure and Applied Logic 79 (1):93-108.

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