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

Author: D. Daigle
Journal: Proc. Amer. Math. Soc. 121 (1994), 1-12
MSC: Primary 13F20; Secondary 13B02
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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