New sufficient condition for the two-dimensional real Jacobian conjecture through the Newton diagram

Abstract

The present paper is devoted to investigating the two-dimensional real Jacobian conjecture. This conjecture claims that if $F=\left(f,g\right):{\mathbb{R}}^2\to {\mathbb{R}}^2$ is a polynomial map with $\det DF\left(x,y\right)\ne 0$ for all $\left(x,y\right)\in {\mathbb{R}}^2$, then $F$ is globally injective. With the help of the Newton diagram, we provide a new sufficient condition such that the two-dimensional real Jacobian conjecture holds. Moreover, this sufficient condition generalizes the main result of [J. Differential Equations {\bf 260} (2016), 5250-5258]. Furthermore, two new classes of polynomial maps satisfying the two-dimensional real Jacobian conjecture are given.