论文标题
关于半代数局部差异性的全球可逆性
On global invertibility of semi-algebraic local diffeomorphisms
论文作者
论文摘要
在本部分说明论文中，我们讨论了$ C^2 $半代数的局部局部差异的全局注射率的条件$ f：\ mathbb {r}^n \ to \ mathbb {r}^n $。在$ n> 2 $的情况下，我们考虑$ \ mathbb {r}^n $的叶子，由$ f $的每个$ n-2 $ phiplactions的级别集，即，映射$ \ m athbb {r}^n \ to \ to \ to \ mathbb {r}^{r}^{r}^{n-2} $ delet delet coilions delet coilions ford coilions ford coilories funcortion fuccortion。众所周知，如果$ f $的非宣传点的一组具有大于或等于$ 2 $的编辑，并且上述叶子的叶子简单地连接了，则$ f $是生物。在这项工作中，我们将这种简单的联系与局部微不足道的纤维概念联系起来。然后，在无穷大的某些可计算的规律条件上确保了这种简单的联系。此外，我们通过使用纤维来提供雅各布猜想的等效陈述。通过示例，我们证明此处介绍的结果与基于光谱假设的先前结果不同。当$ f $由线性同构组成时，我们的考虑也用于讨论某些条件的行为：由于文献中出现了一些误解，这很重要。
In this partly expository paper we discuss conditions for the global injectivity of $C^2$ semi-algebraic local diffeomorphisms $f:\mathbb{R}^n \to \mathbb{R}^n$. In case $n > 2$, we consider the foliations of $\mathbb{R}^n$ defined by the level sets of each $n-2$ projections of $f$, i.e., the maps $\mathbb{R}^n \to \mathbb{R}^{n-2}$ obtained by deleting two coordinate functions of $f$. It is known that if the set of non-proper points of $f$ has codimension greater than or equal to $2$ and the leaves of the above-defined foliations are simply connected, then $f$ is bijective. In this work we relate this simply connectedness with the notion of locally trivial fibrations. Then some computable regularity conditions at infinity ensuring such simply connectedness are presented. Further, we provide an equivalent statement of the Jacobian conjecture by using fibrations. By means of examples we prove that the results presented here are different from a previous result based on a spectral hypothesis. Our considerations are also applied to discuss the behaviour of some conditions when $f$ is composed with linear isomorphisms: this is relevant due to some misunderstandings appearing in the literature.
