On Necessary and Sufficient Conditions for the Real Jacobian Conjecture

Abstract

This paper is an expository one on necessary and sufficient conditions for the real Jacobian conjecture, which states that if $F=\left( f^1,\ldots ,f^n\right) :{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n$ is a polynomial map such that $\det DF\ne 0$, then F is a global injective. In Euclidean space ${\mathbb {R}}^n$, the Hadamard’s theorem asserts that the polynomial map F with $\det DF\ne 0$ is a global injective if and only if $\parallel F\left( {\textbf{x}}\right) \parallel$ approaches to infinite as $\parallel {\textbf{x}}\parallel \rightarrow \infty$. The first part of this paper is to study the two-dimensional real Jacobian conjecture via Bendixson compactification, by which we provide a new version of Sabatini’s result. This version characterizes the global injectivity of polynomial map F by the local analysis of a singular point (at infinity) of a suitable vector field coming from the polynomial map F. Moreover, applying the above results we present a dynamical proof of the two-dimensional Hadamard’s theorem. In the second part, we give an alternative proof of the Cima’s result on the n-dimensional real Jacobian conjecture by the n-dimensional Hadamard’s theorem.