Geometric fields, ranks, and generic derivations

TL;DR

This paper explores minimality and stability conditions in geometric theories of fields, linking properties like stability, simplicity, and rosiness to algebraic and derivation structures, with explicit rank bounds.

Contribution

It establishes new equivalences between model-theoretic properties and algebraic structures in fields with derivations, including conditions for supersimplicity and superrosiness.

Findings

Stable iff strongly minimal in geometric theories of fields

Simple iff SU-rank 1 in such theories

Algebraically bounded stable fields are expansions of algebraically closed fields

Abstract

In this note, we show various minimality results for a geometric theory of fields $T$: $T$ is stable if and only if it is strongly minimal, $T$ is simple if and only if it has SU-rank 1, and $T$ is rosy if and only if $T$ is surgical. Combining the first equivalence with an earlier result of Hrushovski, we deduce that algebraically bounded stable fields are precisely expansions of algebraically closed fields by constants. We then consider algebraically bounded and o-minimal expansions of fields with generic derivations. We show that if $\mathbb{M}$ is a simple algebraically bounded structure and $\Delta$ is a generic tuple of derivations on $\mathbb{M}$, then $(\mathbb{M};\Delta)$ is supersimple if and only if the derivations commute. Similarly, if $\mathbb{M}$ is an o-minimal structure and $\Delta$ is a generic tuple of $T$-derivations on $\mathbb{M}$, then $(\mathbb{M};\Delta)$ is superrosy if and only if the derivations commute. We obtain explicit bounds on ranks using the Kolchin polynomial.