Sufficient conditions for the n-dimensional real Jacobian conjecture

TL;DR

This paper provides new algebraic sufficient conditions for the n-dimensional real Jacobian conjecture, extending previous results to quasi-homogeneous maps and higher dimensions, and offers an alternative proof of a key theorem.

Contribution

It introduces algebraic conditions that ensure injectivity of polynomial maps, generalizing prior work from two to n dimensions and solving an open problem.

Findings

Extended the main result to quasi-homogeneous maps

Generalized the result from 2D to nD

Solved an open problem in the field

Abstract

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].