Bivariables and V\'en\'ereau polynomials

Abstract

We study a family of polynomials introduced by Daigle and Freudenburg, which contains the famous V\'en\'ereau polynomials and defines $\mathbb{A}^2$-fibrations over $\mathbb{A}^2$. According to the Dolgachev-Weisfeiler conjecture, every such fibration should have the structure of a locally trivial $\mathbb{A}^2$-bundle over $\mathbb{A}^2$. We follow an idea of Kaliman and Zaidenberg to show that these fibrations are locally trivial $\mathbb{A}^2$-bundles over the punctured plane, all of the same specific form $X_f$, depending on an element $f\in k[a^{\pm 1},b^{\pm 1}][x]$. 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 V\'en\'ereau 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 $X_f$.