Vénéreau polynomials and related fiber bundles

The Vénéreau polynomialsimagevncolon, equalsy+xn(xz+y(yu+z2)), ngreater-or-equal, slanted1,on image have all fibers isomorphic to the affine space image. Moreover, for all ngreater-or-equal, slanted1 the map image yields a flat family of affine planes over image. In the present note we show that over the punctured plane image, this family is a fiber bundle. This bundle is trivial if and only if vn is a variable of the ring image over image. It is an open question whether v1 and v2 are variables of the polynomial ring image, whereas Vénéreau established that vn is indeed a variable of image over image for ngreater-or-equal, slanted3. In this note we give another proof of Vénéreauʹs result based on the above equivalence. We also discuss some other equivalent properties, as well as˜the relations to the Abhyankar–Sathaye Embedding Problem and to the Dolgachev–Weisfeiler Conjecture on triviality of flat families with fibers affine spaces.