Forking independence in differentially closed fields of positive characteristic
Piotr Kowalski, Omar León Sánchez, Amador Martin-Pizarro
TL;DR
This paper provides a differential-algebraic description of forking independence in $\mathrm{DCF}_{p,m}$, the theory of differentially closed fields of characteristic $p>0$ with $m$ commuting derivations. Central to their approach is the notion of differential trap, a condition on the compositum $KL$ ensuring a controlled differential-algebraic independence beyond algebraic independence; the main result characterizes forking independence as $K \perp^{\mathrm{DCF}}_k L$ if and only if $K$ and $L$ are algebraically independent over $k$ and $KL$ is differentially trap. They also prove that types over algebraically closed subsets of the real sort are stationary, establishing stability, and uncover strongly minimal, strictly disintegrated sets arising from the Bernoulli equation $\partial(T)=T^{p^k+1}$, which are algebraically independent over $\mathbb{F}_p$. The work yields a precise, algebraic handle on forking in positive characteristic differential fields, corrections to earlier claims, and a stability criterion applicable to broader field-operator theories.
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 $\mathbb{F}_p$.