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

Author:
D. Daigle

Journal:
Proc. Amer. Math. Soc. **121** (1994), 1-12

MSC:
Primary 13F20; Secondary 13B02

DOI:
https://doi.org/10.1090/S0002-9939-1994-1227516-X

MathSciNet review:
1227516

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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*.

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© Copyright 1994
American Mathematical Society