Separably differentially closed fields

TL;DR

This paper introduces separably differentially closed fields in positive characteristic, establishing their algebraic and model-theoretic properties, including elementary class status, quantifier elimination, and stability.

Contribution

It defines a new class of differential fields in positive characteristic and proves their fundamental algebraic and model-theoretic properties, including quantifier elimination and stability.

Findings

The class is elementary and characterized by characteristic and differential degree of imperfection.

The theory admits quantifier elimination after adding differential λ-functions.

The class is stable and has prime model extensions.

Abstract

We introduce and study a new class of differential fields in positive characteristic. We call them separably differentially closed fields and demonstrate that they are the differential analogue of separably closed fields. We prove several (algebraic and model-theoretic) properties of this class. Among other things, we show that it is an elementary class, whose theory we denote $\SDCF$, and that its completions are determined by specifying the characteristic $p$ and the differential degree of imperfection $\epsilon$. Furthermore, after adding what we call the differential $\lambda$-functions, we prove that the theory $\SDCFl$ admits quantifier elimination, is stable, and prime model extensions exist.