论文标题
真正的雅各布猜想的必要条件
The necessary and sufficient conditions for the real Jacobian conjecture
论文作者
论文摘要
真正的雅各布猜想声称，如果$ f = \ left（f^1，\ ldots，f^n \ right）：\ mathbb {r}^n \ rightArrow \ rightArrow \ mathbb {r}^n $是一张多项式地图，那么$ \ det df $ y y y y y y y y y y y y n of $ f $ a $ f $ a $ f $ a Inment in Inment in Inment in Inment in Inverive。 第一部分是通过动态系统定性理论的方法来研究二维实际雅各布猜想。通过Bendixson紧凑型，可以从与多项式MAP $ F $相关的Hamiltonian系统获得诱导的多项式差异系统。我们证明以下语句是等效的：（a）$ f $是全球注射剂； （b）诱导系统的起源是中心； （c）诱导系统的起源是单粒子奇异点； （d）诱导系统的起源没有双曲线扇区； （e）诱导系统具有$ c^k $的第一积分，其原始位于孤立的minimun和$ k \ in \ mathbb {n}^{+} \ cup \ {\ cup \ {\ infty \} $。此外，应用上述结果我们提出了二维实际雅各布猜想的有效性的必要条件，这是代数标准。根据定义，当$ f $且仅当标准函数的限制是无限为$ \ weft | x \ right |+\ weft | y \ right | $时，$ f $是一个全局注射剂。该代数标准改善了Braun等人的主要结果[J.微分方程{\ bf 260}（2016）5250-5258]。 在第二部分中，获得了$ n $二维的实际雅各布猜想的必要条件。 Using the tool from the nonlinear functional analysis, $F$ is a global injective if and only if $\parallel F\left(\mathbf{x}\right)\parallel$ approaches to infinite as $\parallel\mathbf{x}\parallel\rightarrow\infty$, which is a generalization of the above algebraic criterion.作为一个应用程序，我们在$ n $ dimensional的真实Jacobian猜想[nonlinear肛门上提供了CIMA结果的替代证明。 {\ bf 26}（1996）877-885]。
The real Jacobian conjecture claims that if $F=\left(f^1,\ldots,f^n\right):\mathbb{R}^n\rightarrow \mathbb{R}^n$ is a polynomial map such that $\det DF$ is nowhere zero, then $F$ is a global injective. The first part is to study the two-dimensional real Jacobian conjecture via the method of the qualitative theory of dynamical systems. By Bendixson compactification, an induced polynomial differential system can be obtained from the Hamiltonian system associated to polynomial map $F$. We prove that the following statements are equivalent: (A) $F$ is a global injective; (B) the origin of induced system is a center; (C) the origin of induced system is a monodromic singular point; (D) the origin of induced system has no hyperbolic sectors; (E) induced system has a $C^k$ first integral with an isolated minimun at the origin and $k\in\mathbb{N}^{+}\cup\{\infty\}$. Moreover, applying the above results we present a necessary and sufficient condition for the validity of the two-dimensional real Jacobian conjecture, which is an algebraic criterion. By definition a criterion function, $F$ is a global injective if and only if the limit of criterion function is infinite as $\left|x\right|+\left|y\right|$ tends to infinity. This algebraic criterion improves the main result of Braun et al [J. Differential Equations {\bf 260} (2016) 5250-5258]. In the second part, the necessary and sufficient conditions on the $n$-dimensional real Jacobian conjecture is obtained. Using the tool from the nonlinear functional analysis, $F$ is a global injective if and only if $\parallel F\left(\mathbf{x}\right)\parallel$ approaches to infinite as $\parallel\mathbf{x}\parallel\rightarrow\infty$, which is a generalization of the above algebraic criterion. As an application, we give an alternate proof of the Cima's result on the $n$-dimensional real Jacobian conjecture [Nonlinear Anal. {\bf 26} (1996) 877-885].
