Model theory of derivations of the Frobenius map revisited

TL;DR

This paper advances the model theory of fields with Frobenius derivations by proving axiomatizability, quantifier elimination, and providing a new geometric axiomatization, thereby strengthening and generalizing previous results.

Contribution

It establishes the axiomatizability and quantifier elimination of the model companion for fields with Frobenius derivations, extending prior work and introducing a geometric axiomatization.

Findings

Model companion is axiomatizable using Wood's axioms.

Quantifier elimination is achieved after adding Frobenius inverse.

Provides a new geometric axiomatization of the model companion.

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.