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Preprint Number 788
788. Pablo Cubides-Kovacsics and Françoise Delon
Definable types in algebraically closed valued fields
Submission date: 14 OCtober 2014.
Marker and Steinhorn shown that given two models M\prec N of an o-minimal theory, if all 1-types over M realized in N are definable, then all types over M realized in N are definable. In this article we characterize pairs of algebraically closed valued fields satisfying the same property. Although it is true that if M is an algebraically closed valued field such that all 1-types over M are definable then all types over M definable, we build a counterexample for the relative statement, i.e., we show for any n\geq 1 that there is a pair M\prec N of algebraically closed valued fields such that all n-types over M realized in N are definable but there is an n+1-type over M realized in N which is not definable. Finally, we discuss what happens in the more general context of C-minimality.
Mathematics Subject Classification: Primary 12J10, 03C60, 13L05, Secondary 03C98
Keywords and phrases: Algebraically closed valued fields, definable types
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