Model Theory of Derivations of the Frobenius Map Revisited

Abstract

We prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory $\operatorname {DCF}_p$ and that it eliminates quantifiers after adding the inverse of the Frobenius map to the language. This strengthens the results from [4]. As a by-product, we get a new geometric axiomatization of this model companion. Along the way we also prove a quantifier elimination result, which holds in a much more general context and we suggest a way of giving “one-dimensional” axiomatizations for model companions of some theories of fields with operators.

Operators coming from ring schemes

“…Remark For $$\mathcal {B}=B_\otimes$$, adding just the inverse of the Frobenius map suffices to get quantifier elimination for $$\mathcal {B}-\operatorname{CF}$$, which was proven in [1]. During the review process of this paper, the first author showed quantifier elimination results regarding the language with the inverse of the Frobenius map, which can be applied to the theories $$\mathcal {B}-\operatorname{CF}$$ (see [7, Theorem 2.8])…”