Analogs of the Shapiro Shapiro Conjecture in Positive Characteristic

TL;DR

This paper investigates conditions under which rational functions with rational ramification points are equivalent to those defined over the base field, focusing on fields of characteristic 2 and 3, and exploring limitations in higher characteristics.

Contribution

It provides new results characterizing when such rational functions are equivalent to those over the base field in characteristic 2 and 3, and discusses limitations in higher characteristics.

Findings

Characterizes conditions in characteristic 2 and 3

Shows insufficiency of natural conditions in higher characteristic

Provides criteria for rational function equivalence

Abstract

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.