The definable multiplicity property and generic automorphisms

Abstract

Let $\text{T}$ be a strongly minimal theory with quantifier elimination. We show that the class of existentially closed models of $\text{T∪{“σ}$ is an automorphism”} is an elementary class if and only if $\text{T}$ has the definable multiplicity property, as long as $\text{T}$ is a finite cover of a strongly minimal theory which does have the definable multiplicity property. We obtain cleaner results working with several automorphisms, and prove: the class of existentially closed models of $\text{T∪{“σ}{}_{\mathrm{i}}$ is an automorphism”: $\text{i=1,2}}$ is an elementary class if and only if $\text{T}$ has the definable multiplicity property.