Stationarity and elimination of imaginaries in stable and simple theories

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 fields. 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.