Vénéreau-type polynomials as potential counterexamples

We study some properties of the Vénéreau polynomials bm=y+xm(xz+y(yu+z2))∈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≥3, they are C[x]-coordinates. For m=1,2, it is only known that they are 1-stable C[x]-coordinates. We show that b2 is in fact a 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 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 C[x]-coordinates.