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

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 A is irreducible for each A E k and such that Rp C k[u, v] for some v E R and some integer n > 0. Given a weak variable u of R, consider all v E R such that Rp C k[u, v] for some n; if one of these v is "absolutely smaller" than u (roughly, degx v 0 and R is a polynomial ring in two variables over k (which is abbreviated R = k[2]) . A variable of R is an element u of R such that R = k[u, v] for some v E R; the set of variables of R is denoted Var(R) . A coordinate system of R is an ordered pair (u, v) E R x R such that R = k[u, v]; the set of coordinate systems of R is denoted F(R). One can safely assert that little is known about the subalgebras A = k[2] of R which satisfy Rp C A for n large enough-the set of all such A 's is denoted ?(R) . For instance, the question whether every u E R which satisfies R/uR = k1l] is a variable of such a subalgebra A is still unanswered, after at least twelve years of existence (this is a weak version of the conjecture discussed in [7]). Questions like this one would be easier if we understood how the variables of these subalgebras look, as elements of R. We define weak variable of R to mean an element u of R which is a variable of some A E ?(R) and which is such that u A is irreducible in R for every A E k. A weak variable is proper if it is not a variable of R. Given a weak variable u of R, we discuss the following two questions: (1) Can we describe the set Xu(R) = {A E X(R)Iu E Var(A)}? The first section shows, in particular, that if u ? k[RP] (which is always the case if k is perfect) then that collection of rings is totally ordered by inclusion and, consequently, has a unique maximal element. The second section is concerned with: Received by the editors June 8, 1992. 1991 Mathematics Subject Classification. Primary 13F20. The author was supported by a grant from NSERC Canada. (g 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 04:34:08 UTC All use subject to http://about.jstor.org/terms