On existentially closed partial differential fields with an automorphism Les modèles existentiellement clos de la théorie des corps à dérivées partielles munis d’un automorphisme

In this paper, we study the existentially closed models of the theory of partial differential fields (of characteristic zero) with an automorphism. These models are called partial differentially closed fields with an automorphism or, shorter, PDCFA. We first give a characterization of PDCFA’s in terms of differential varieties and then use this latter to obtain some algebraic properties of these fields. In particular we prove a Nullstellensatz theorem and some interesting results on two particular subfields of a PDCFA (namely the subfield of constants and the fixed field). All the results presented here generalize the one in the work of R. Bustamente in the case of difference differential fields, that is fields equipped with a single derivation and an automorphism which commute ([Bu]).