Bivariables and Vénéreau polynomials

We study a family of polynomials introduced by Daigle and Freudenburg, which contains the famous Vénéreau polynomials and defines 𝔸 2 -fibrations over 𝔸 2 . According to the Dolgachev–Weisfeiler conjecture, every such fibration should have the structure of a locally trivial 𝔸 2 -bundle over 𝔸 2 . We follow an idea of Kaliman and Zaidenberg to show that these fibrations are locally trivial 𝔸 2 -bundles over the punctured plane, all of the same specific form X f , depending on an element f∈k[a ±1 ,b ±1 ][x]. We then introduce the notion of bivariables and show that the set of bivariables is in bijection with the set of locally trivial bundles X f that are trivial. This allows us to give another proof of Lewis’s result stating that the second Vénéreau polynomial is a variable and also to trivialise other elements of the family X f . We hope that the terminology and methods developed here may lead to future study of the whole family X f .