A structural approach to diophantine definability
Mihai Prunescu · KOPS (University of Konstanz) · 1998
General and diophantine definability in number rings and their polynomial rings are studied from a model-theoretic point of view. The main tool used is a modern form of the Theorem of Beth. For a ring R we consider the monoid EndR (R ) of all embeddings of a nonstandard enlargement R in itself which fix the standard elements. If R is a number ring or any field, the application of natural restriction ResR from End R[T ] (R[T ] ) to EndR (R ) is a well defined homomorphism of monoids. We give connections between the diophantine definability of the integers Z in a number ring R, a phenomenon of transfer of definability from the polynomial ring R[T ] to the ring R, and properties of the homomorphism ResR . In the case of the ring Z itself we get as a byproduct that the restriction ResZ : End Z[T ] (Z[T ] ) ~ \\Gamma! EndZ (Z ) is an isomorphism of monoids, fact which is equivalent with the combination of two well known theorems of Y. Matiyasevich and J. Denef. We pr...