Jacobian conjecture in $\mathbb R^2$

TL;DR

This paper proves the Jacobian conjecture for polynomial maps in two-dimensional real space using dynamical systems theory, confirming injectivity when the Jacobian determinant is a nonzero constant.

Contribution

It provides a proof of the Jacobian conjecture in -dimensional real space, a longstanding open problem, by applying dynamical systems techniques.

Findings

Confirmed injectivity of polynomial maps in  with constant nonzero Jacobian

Extended understanding of the Jacobian conjecture in real two-dimensional space

Utilized dynamical systems methods to solve a classical algebraic problem

Abstract

Jacobian conjecture states that if $F:\ \mathbb C^n(\mathbb R^n)\rightarrow \mathbb C^n(\mathbb R^n)$ is a polynomial map such that the Jacobian of $F$ is a nonzero constant, then $F$ is injective. This conjecture is still open for all $n\ge 2$, and for both $\mathbb C^n$ and $\mathbb R^n$. Here we provide a positive answer to the Jacobian conjecture in $\mathbb R^2$ via the tools from the theory of dynamical systems.