On Necessary and Sufficient Conditions for the Real Jacobian Conjecture.

This paper is an expository one on necessary and sufficient conditions for the real Jacobian conjecture, which states that if F = f 1 , … , f n : R n → R n is a polynomial map such that det D F ≠ 0 , then F is a global injective. In Euclidean space R n , the Hadamard’s theorem asserts that the polynomial map F with det D F ≠ 0 is a global injective if and only if ‖ F x ‖ approaches to infinite as ‖ x ‖ → ∞ . 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. [ABSTRACT FROM AUTHOR]