On the model companion of partial differential fields with an automorphism

Abstract

We prove that the class of partial differential fields of characteristic zero with an automorphism has a model companion. We then establish the basic model theoretic properties of this theory and prove that it satisfies the canonical base property, and consequently the Zilber dichotomy, for finite-dimensional types.

The Model Companion of Differential Fields With Free Operators

“…(c) In the pure D-fields case when ∆ = ∅, the prolongations introduced here coincide with the D-prolongations introduced in [6] and used in [7]. (d) When D(R) = R × R, the prolongation introduced here is the differencedifferential prolongation V × V σ that was used in [5].…”

“…As in the case of existentially closed difference-differential fields considered in [5], we do not know if the above characterisation amounts to a first-order axiomatisation because in differentially closed fields the property of being Kolchin irreducible is not known to be a definable property of the parameters of a ∆-closed set. Our strategy to avoid this difficulty is precisely the same as that of the first author in [5].…”

“…For the latter the main point is that, by a criterion of Rosenfeld's, being the characteristic set of a prime ∆-ideal is definable in parameters. See [5] for a more detailed discussion.…”

“…The existence of the model companion for this theory was established by the first author in [5], and served as a model for us.…”

“…Because irreducibility of Kolchin closed sets is not known to be a definable property of the parameters, one has to modify the characterisation into a provably first-order axiomatisation of the existentially closed models. Here we follow the method established by the first author in [4] and [5], which uses characteristic sets of prime differential ideals.…”