Sufficient conditions for the n-dimensional real Jacobian conjecture

The real Jacobian conjecture was posed by Randall in 1983. This conjecture asserts that if $F=\left(f_1,\ldots ,f_n\right):\mathbb{R}^n\rightarrow\mathbb{R}^n$ is a polynomial map such that $\det DF\left(\mathbf{x}\right)\neq0$ for all $\mathbf{x}\in\mathbb{R}^n$, then $F$ is injective. This investigation mainly consists of two parts. Firstly, we use the qualitative theory of dynamical systems to give an alternate proof of the polynomial version of the $n$-dimensional Hadamard's theorem. Secondly, we present some algebraic sufficient conditions for the $n$-dimensional real Jacobian conjecture. Our results not only extend the main result of [J. Differential Equations {\bf 260} (2016), 5250-5258] to quasi-homogeneous type, but also generalize it from $\mathbb{R}^2$ to $\mathbb{R}^n$. As a coproduct of our proof process, we solve an open problem formulated by Braun, Gin\'{e} and Llibre in [J. Differential Equations {\bf 260} (2016), 5250-5258].