Mathematics – Commutative Algebra
Scientific paper
2010-07-14
Mathematics
Commutative Algebra
13 pages; Generalized results of previous version to Venereau-type polynomials and added proposition 3
Scientific paper
We study some properties of the Venereau polynomials f_n=y+x^n(xz+y(yu+z^2)), a sequence of proposed counterexamples to the Abhyankar-Sathaye embedding conjecture. It is well known that these are hyperplanes, and for n at least 3, they are C[x]-coordinates. For n=1,2, it is known that they become coordinates upon going modulo x or upon inverting x, and that they are 1-stable C[x]-coordinates. We show that f_2 is in fact a C[x]-coordinate. We introduce the notion of Venereau-type polynomials, and show that these are all hyperplanes, coordinates upon going modulo x or upon inverting x, and stably tame, 1-stable C[x]-coordinates. We show that some of these Venereau-type polynomials are in fact C[x]-coordinates; the rest remain potential counterexamples to the embedding and other conjectures. For those that we show to be coordinates, we also show that any automorphism with one of them as a component is stably tame.
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