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Diophantine equivalence and countable rings

Published online by Cambridge University Press:  12 March 2014

Alexandra Shlapentokh
Alexandra Shlapentokh*
Affiliation:
Department of Mathematics, East Carolina University, Greenville, North Carolina 27858-4353, E-mail: mashlape@ecuvax.cis.ecu.edu
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Abstract

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 holomorphy rings of ℚ.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 3 , September 1994 , pp. 1068 - 1095
Copyright
Copyright © Association for Symbolic Logic 1994

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