Forking independence in differentially closed fields of positive characteristic

TL;DR

This paper characterizes forking independence in differentially closed fields of positive characteristic using differential algebra, and explores the properties of solutions to specific Bernoulli differential equations.

Contribution

It provides a differential-algebraic framework for understanding forking independence in DCF$_{p,m}$ and analyzes the model-theoretic properties of certain differential equations.

Findings

Types over algebraically closed sets are stationary.

Solutions to the Bernoulli differential equation are strongly minimal.

The solution set is algebraically independent over _p.

Abstract

We provide a differential-algebraic description of forking independence in the stable theory DCF$_{p,m}$ of differentially closed fields of characteristic $p>0$ with $m$-many commuting derivations. As a by-product of this description, we prove that types over algebraically closed subsets of the real sort are stationary. In addition, we prove that the set of non-zero solutions to the Bernoulli differential equation $x'=x^{p^k+1}$ with $k>0$ is strongly minimal and its geometry is strictly disintegrated, which implies that this set is algebraically independent over $\mathbb{F}_p$.