Fields with Lie-commuting and iterative operators

We introduce a general framework for studying fields equipped with operators, given as co-ordinate functions of homomorphisms into a local algebra D, satisfying various compatibility conditions that we denote by Γ and call such structures D^Γ-fields. These include Lie-commutativity of derivations and g-iterativity of (truncated) Hasse-Schmidt derivations. Our main result is about the existence of principal realisations of D^Γ-kernels. As an application, we prove companionability of the theory of D^Γ-fields and denote the companion by D^Γ-CF. In characteristic zero, we prove that D^Γ-CF is a stable theory that satisfies the CBP and Zilber's dichotomy for finite-dimensional types. We also prove that there is a uniform companion for model-complete theories of large D^Γ-fields, which leads to the notion of D^Γ-large fields and we further use this to show that PAC substructures of D^Γ-DCF are elementary.