Charlotte Bartnick | Searcharxiv

arXiv2025

Embedding stable groups into algebraic groups

Adapting a proof of Bouscaren and Delon, we show that every type-definable connected group in a given stable theory of fields embeds into an algebraic group, under a condition on the definable closure. We also present general hypotheses which yield a uniform description of the definable closure in such theories of fields. The setting includes in particular the theories of separably closed fields of arbitrary degree of imperfection and differentially closed fields of arbitrary characteristic.

math.LO↗

arXiv2025

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 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.

math.LO↗