The necessary and sufficient conditions for the real Jacobian conjecture

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].