Purely inseparable extensions of $\textbf {k}[X,Y]$

Abstract:

Let k be a field of characteristic $p > 0$ and R a polynomial ring in two variables over k. Define weak variable of R to mean an element u of R such that $u - \lambda$ is irreducible for each $\lambda \in {\mathbf {k}}$ and such that ${R^{{p^n}}} \subseteq {\mathbf {k}}[u,v]$ for some $v \in R$ and some integer $n \geq 0$. Given a weak variable u of R, consider all $v \in R$ such that ${R^{{p^n}}} \subseteq {\mathbf {k}}[u,v]$ for some n; if one of these v is "absolutely smaller" than u (roughly, ${\deg _X}v < {\deg _X}u$ for all coordinate systems (X, Y) of R), we call it an R-companion of u. The main result gives a connection between the structure of a purely inseparable extension $R \supset A$, where A is a polynomial ring in two variables, and whether or not there exists a companion for each u in a suitable set of weak variables of R.