Résume | All fields under discussion here are assumed to have finite characteristic p. This talk might be seen as a sequel to my survey talk at Françoise Delon's conference in June 2016, although it will not assume familiarity with this talk.
Of interest here are two complete theories, namely differentially closed fields (DCF) and separably closed fields (inf-SCF) with infinite degree of imperfection. These theories are related. For example, the underlying field of a model of DCF is a model of inf-SCF, and the constant field is also a model of inf-SCF. In each case, there are natural choices of language in which the theory has quantifier elimination.
We will consider ways in which the theories are not alike. In the mid 1980's Gabriel Srour proved that DCF is equational, and also that the theories of separably closed fields of finite degree of imperfection are equational. However, to my knowledge, the equationality of inf-SCF is still unknown.
Delon proved that the finite imperfection separably closed fields have elimination of imaginaries (EI); this too is an open question for inf-SCF.
At the June conference, Zoé Chatzidakis and Silvain Rideau asked whether DCF might have EI.
Upon reflection and with a bit of work, I realized that the answer is yes. The proof involves an idea which Srour used in his proof of equationality for DCF. After providing some necessary background about DCF and inf-SCF, I will describe this recent work. |