Model Companions of TAut for Stable T

Let T be a complete first order theory in a countable relational language L. We assume relation symbols have been added to make each formula equivalent to a predicate. Adjoin a new unary function symbol σ to obtain the language Lσ; TAut is obtained by adding axioms asserting that σ is an L-automorphism. The modern study of the model companion of theories with an automorphism has two aspects. One line, stemming from Lascar [7], deals with ‘generic’ automorphisms of arbitrary structures. A second, beginning with Chatzidakis and Hrushovski [2] and questions of Macintyre about the Frobenius automorphism is more concerned with specific algebraic theories. This paper is more in the first tradition: we find general necessary and sufficient conditions for a stable first order theory with automorphism to have a model companion. Kikyo investigates the existence of model companions of TAut when T is unstable in [4]. He also includes an argument of Kudaibergenov showing