TITLE: Point-line incidence bounds over positive characteristic fields
QUESTION [6 upvotes]: I am aware of work on point-line incidence bounds over $\mathbb{R}$, $\mathbb{C}$, and finite fields, in particular various versions of the Szemeredi-Trotter bounds. I would like to know if work along these lines has been done over function fields like $\mathbb{F}_p(t)$. I looked around some and asked some people, but I didn't find anything.
What I'm actually really interested in is arbitrary fields. Results on $\mathbb{C}$ often generalize to arbitrary characteristic zero fields, and results on finite fields imply results on algebraically closed fields of characteristic $p$. So that's why I ask about $\mathbb{F}_p(t)$.

REPLY [5 votes]: The current best Szemeredi-Trotter bound over finite fields also holds over arbitrary fields. It is in the wonderful paper "An Improved Point-Line Incidence Bound Over Arbitrary Fields" by Sophie Stevens and Frank de Zeeuw.
Unfortunately, this current best bound of $m^{11/15}n^{11/25}$ is closer to the trivial $m^{3/4}n^{3/4}$ than to the conjectured $m^{2/3}n^{2/3}$.
In a somewhat different direction, model theorists generalize these bounds to things such as o-minimal frameworks and distal. See Chernikov, Galvin, and Starchenko
 and also Basu and Raz