A property of polynomial curves over a field of positive characteristic

Abstract:

Let ${\mathbf {k}}$ be an algebraically closed field of characteristic $p > 0$. We show that if $F \in {\mathbf {k}}\left [ {X,Y} \right ]$ is a rational curve with one place at infinity and with nonprincipal bidegree, such that $\theta \left ( F \right ) \in {\mathbf {k}}\left [ {{X^p},Y} \right ]$ for some automorphism $\theta$ of ${\mathbf {k}}\left [ {X,Y} \right ]$, then $\theta$ can be chosen to be either linear or of "de Jonquière" type. We also give consequences of that fact for the problem of classifying the embeddings of the line in the plane.