Vénéreau-type polynomials as potential counterexamples

Abstract

We study some properties of the Vénéreau polynomials $b_m=y+x^m(xz+y(yu+z^2)) \in \mathbb{C}[x,y,z,u]$, a sequence of proposed counterexamples to the Abhyankar-Sathaye embedding conjecture and the Dolgachev-Weisfeiler conjecture. It is well known that these are hyperplanes and residual coordinates, and for $m \geq 3$, they are $\mathbb{C}[x]$-coordinates. For $m=1,2$, it is only known that they are 1-stable $\mathbb{C}[x]$-coordinates. We show that $b_2$ is in fact a $\mathbb{C}[x]$-coordinate. We introduce the notion of Vénéreau-type polynomials, and show that these are all hyperplanes and residual coordinates. We show that some of these Vénéreau-type polynomials are in fact $\mathbb{C}[x]$-coordinates; the rest remain potential counterexamples to the aforementioned 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. The remainder are stably tame, 1-stable $\mathbb{C}[x]$-coordinates.

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