Stationarity and elimination of imaginaries in stable and simple theories
Charlotte Bartnick
TL;DR
The paper develops a general framework in which a stable base theory $T_0$ with elimination of quantifiers and imaginaries controls a simple expansion $T$, ensuring stationarity of types over real algebraically closed sets when $T$ is appropriately 'controlled' by $T_0$. It leverages Wagner’s theory of definable hulls in stable groups to derive a criterion that yields failure of geometric elimination of imaginaries for several prominent theories, including beautiful pairs $T_0P$ and the theory of separably closed fields of infinite degree of imperfection $SCF_p^{\infty}$, even after naming submodels. The results unify and extend classical non-elimination proofs (Delon; Messmer–Wood; Pillay–Vassiliev) within a single, model-theoretic framework and apply to new examples such as separably differentially closed fields. The work provides both a robust mechanism to detect non-elimination of imaginaries and a broad set of stationary-type results, clarifying the interplay between stability, imaginaries, and expansions by field-like structures.
Abstract
We show that types over real algebraically closed sets are stationary, both for the theory of separably closed fields of infinite degree of imperfection and for the theory of beautiful pairs of algebraically closed field. The proof is given in a general setup without using specific features of theories of fields. Moreover, we generalize results of Delon as well as of Messmer and Wood that separably closed fields of infinite degree of imperfection and differentially closed fields of positive characteristic do not have elimination of imaginaries. Using work of Wagner on subgroups of stable groups, we obtain a general criterion yielding the failure of geometric elimination of imaginaries. This criterion applies in particular to beautiful pairs of algebraically closed fields, giving an alternative proof of the corresponding result of Pillay and Vassiliev.