We obtain a new theorem for the non-properness set $S_f$ of a non-singular polynomial mapping $f:\mathbb C^n \to \mathbb C^n$. In particular, our result shows that if f is a counterexample to the Jacobian conjecture, then $S_f\cap Z \neq \emptyset $, for every hypersurface Z dominated by $\mathbb C^{n-1}$ on which some non-singular polynomial $h: \mathbb C^{n}\to \mathbb C$ is constant. Also, we present topological approaches to the Jacobian conjecture in $\mathbb C^n$. As applications, we extend bidimensional results of Rabier, Lê and Weber to higher dimensions.

中文翻译：

---

关于ℂn中雅可比猜想的拓扑方法

我们得到了一个关于非属性集的新定理$S_f$非奇异多项式映射的$f:\mathbb C^n \to \mathbb C^n$. 特别是，我们的结果表明，如果F是雅可比猜想的反例，则$S_f\cap Z \neq \emptyset $, 对于每个超曲面Z被）占据$\mathbb C^{n-1}$其中一些非奇异多项式$h: \mathbb C^{n}\to \mathbb C$是恒定的。此外，我们提出了雅可比猜想的拓扑方法$\mathbb C^n$. 作为应用，我们将 Rabier、Lê 和 Weber 的二维结果扩展到更高维度。