Analogs of the Shapiro Shapiro Conjecture in Positive Characteristic

Motivated by the Shapiro Shapiro conjecture, we consider the following: given a field $k$, under what conditions must a rational function with only $k$-rational ramification points be equivalent (after post-composition with a fractional linear transformation) to a rational function defined over $k$? The main results of this paper answer this question when $k$ has characteristic 2 or 3. We also show the insufficiency of several natural conditions in higher characteristic.