The definable multiplicity property and generic automorphisms
Let \(T\) be a complete theory with quantifier elimination in a language \(L\) and let \(T_{\sigma}\) denote the (possibly incomplete) expansion \(T \cup \{ \sigma\) is an automorphism\} in the suitable language extending \(L\) by a new 1-ary function symbol. A well-known example in this setting concerns \(T = \text{ACF}\), the theory of algebraically closed fields (with a fixed characteristic to ensure completeness); \(\text{ACF}_{\sigma}\) has a model companion, called ACFA and admitting a nice axiomatization. For an arbitrary \(T\), one may ask under which conditions \(T_{\sigma}\) has a model companion. Here the authors deal with this problem when \(T\) is strongly minimal (so ACF lies in this framework). The leading conjecture is that, under the strongly minimal assumption, a model companion exists if and only if \(T\) satisfies the definable multiplicity property DMP (a condition essentially requiring the definability of the Morley degree). The implication from right to left is known, and the authors show here the converse implication when \(T\) is a finite cover of a strongly minimal theory which does have the DMP, in particular when \(T\) is locally modular. Notably, nicer results are obtained when working with \(\geq 2\) automorphisms \(\sigma_i\) \((i \in I)\). It turns out that, in this case, for every strongly minimal \(T\), \(T \cup \{\sigma_i\) is an automorphism\(: i \in I \}\) has a model companion exactly when \(T\) has the DMP. In particular, for a given strongly minimal \(T\), the existence of a model companion for the expansion of \(T\) by 2 automorphisms implies the existence of a model companion when infinitely many automorphisms occur.
scientific article; zbMATH DE number 1546604