Definable types in algebraically closed valued fields |
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| Authors: | Pablo Cubides Kovacsics Françoise Delon |
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| Institution: | 1. Laboratoire Paul Painlevé (Unité Mixte de Recherche 8524), Université de Lille et Centre National de Recherche Scientifique, Villeneuve d'Ascq, France;2. équipe de Logique Mathématique, Institut de Mathématiques de Jussieu‐Paris Rive Gauche (Unité Mixte de Recherche 7586), Université Paris Diderot et Centre National de Recherche Scientifique, Unité de Formation et de Recherche de Mathématiques, Paris 13, France |
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| Abstract: | In 15 , Marker and Steinhorn characterized models  of an o‐minimal theory such that all types over M realized in N are definable. In this article we characterize pairs of algebraically closed valued fields satisfying the same property. In o‐minimal theories, a pair of models  for which all 1‐types over M realized in N are definable has already the desired 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 are definable, we build a counterexample for the relative statement, i.e., we show for any  that there is a pair  of algebraically closed valued fields such that all n‐types over M realized in N are definable but there is an  ‐type over M realized in N which is not definable. |
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