论文标题
双变量和vénéreau多项式
Bivariables and Vénéreau polynomials
论文作者
论文摘要
我们研究了Daigle和Freudenburg引入的多项式家庭，其中包含著名的Vénéreau多项式，并定义了$ \ Mathbb {a}^2 $ - 2 $ - 纤维，超过$ \ Mathbb {A a}^2 $。根据Dolgachev-Weisfeiler的猜想，每个这样的振动都应具有本地琐碎的$ \ mathbb {a}^2 $ -Bundle $ \ Mathbb {a}^2 $的结构。我们遵循Kaliman和Zaidenberg的想法，以表明这些纤维在局部是微不足道的$ \ Mathbb {a}^2 $ - 在刺穿的平面上捆绑，所有相同的特定表格$ x_f $，取决于k [a^{a^{a^{\ pm pm^}，b^，b^，b^，b^，b^，b^{\ pm 1} $ f \ in然后，我们介绍了双变量的概念，并表明双变量的集合与一组本地琐碎的捆绑包$ x_f $相同。这使我们能够给出另一个刘易斯结果的证明，表明第二个vénéreau多项式是一个变量，并且也使家庭的其他元素琐碎$ x_f $。我们希望这里开发的术语和方法可能会导致对整个家庭的未来研究$ x_f $。
We study a family of polynomials introduced by Daigle and Freudenburg, which contains the famous Vénéreau 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 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 $X_f$.
