Model theory of fields with free operators in positive characteristic

TL;DR

This paper characterizes when the theory of fields with certain algebraic operators in positive characteristic admits a model companion, showing it is strictly stable and admits quantifier elimination in a minimal language.

Contribution

It provides algebraic criteria for the existence of model companions of fields with $B$-operators in positive characteristic, including stability and quantifier elimination results.

Findings

Model companions exist under specific algebraic conditions.

Theories are strictly stable with quantifier elimination.

Forking relation is explicitly described.

Abstract

We give algebraic conditions about a finite algebra $B$ over a perfect field of positive characteristic, which are equivalent to the companionability of the theory of fields with "$B$-operators" (i.e. the operators coming from homomorphisms into tensor products with $B$). We show that, in the most interesting case of a local $B$, these model companions admit quantifier elimination in the "smallest possible" language and they are strictly stable. We also describe the forking relation there.