Sufficient conditions for the n-dimensional real Jacobian conjecture

The real Jacobian conjecture, proposed by Randall in 1983, asserts that a polynomial map $F=(f_1,\dots ,f_n):R^n\to R^n$ such that $\det DF\left(x\right)\ne 0$ for all $x\in R^n$ is injective. However, this conjecture is disproven by Pinchuk's counterexample.

This investigation mainly consists of two parts. Firstly, we use the qualitative theory of dynamical systems to give an alternative 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 260 (2016), 5250-5258] to quasi-homogeneous type, but also generalize it from $R^2$ to $R^n$. As a coproduct of our proof process, we solve an open problem formulated by Braun, Giné and Llibre in [J. Differential Equations 260 (2016), 5250-5258].