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The real Jacobian conjecture claims that if$F=\left(f^1,\ldots,f^n\right):\mathbb{R}^n\rightarrow \mathbb{R}^n$ is apolynomial map such that $\det DF$ is nowhere zero, then $F$ is a globalinjective.  The first part is to study the two-dimensional real Jacobian conjecture viathe method of the qualitative theory of dynamical systems. By Bendixsoncompactification, an induced polynomial differential system can be obtainedfrom the Hamiltonian system associated to polynomial map $F$. We prove that thefollowing statements are equivalent: (A) $F$ is a global injective; (B) theorigin of induced system is a center; (C) the origin of induced system is amonodromic singular point; (D) the origin of induced system has no hyperbolicsectors; (E) induced system has a $C^k$ first integral with an isolated minimunat the origin and $k\in\mathbb{N}^{+}\cup\{\infty\}$. Moreover, applying theabove results we present a necessary and sufficient condition for the validityof the two-dimensional real Jacobian conjecture, which is an algebraiccriterion. By definition a criterion function, $F$ is a global injective if andonly if the limit of criterion function is infinite as$\left|x\right|+\left|y\right|$ tends to infinity. This algebraic criterionimproves 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 thenonlinear 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 ofthe above algebraic criterion. As an application, we give an alternate proof ofthe Cima's result on the $n$-dimensional real Jacobian conjecture [NonlinearAnal. {\bf 26} (1996) 877-885].

The necessary and sufficient conditions for the real Jacobian conjecture