A self-map F of an affine space ${\bf A}_k^n $ over a field k is said to be a Keller map if F is given by polynomials F1, …, Fn ∈ k[X1, …, Xn] whose Jacobian determinant lies in $k\setminus \{0\}$. We consider char(k) = 0 and assume, as we may, that the Fis vanish at the origin. In this note, we prove that if F is Keller then its base ideal IF = 〈F1, …, Fn〉 is radical (a finite intersection of maximal ideals in this case). We then conjecture that IF = 〈X1, …, Xn〉, which we show to be equivalent to the classical Jacobian Conjecture. In addition, among other remarks, we observe that every complex Keller map admits a well-defined multidimensional global residue function.

中文翻译：

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凯勒图的理想理论方法

自我地图F仿射空间${\bf A}_k^n $在一个领域ķ被称为凯勒图，如果F由多项式给出F1, …,Fn∈ķ[X1, …,Xn] 其雅可比行列式位于$k\setminus \{0\}$. 我们考虑 char(ķ) = 0 并且我们可以假设F一世s 消失在原点。在本笔记中，我们证明如果F是凯勒，那么它的基本理想一世F= 〈F1, …,Fn〉是激进的（在这种情况下是最大理想的有限交集）。然后我们推测一世F= 〈X1, …,Xn〉，我们证明它等价于经典的雅可比猜想。此外，在其他评论中，我们观察到每个复杂的 Keller 映射都承认一个定义明确的多维全局残差函数。