Jacobian conjecture in ${\mathbb{R}}^2$

Abstract

Jacobian conjecture states that if $F:{\mathbb{C}}^n({\mathbb{R}}^n)\to {\mathbb{C}}^n({\mathbb{R}}^n)$ 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.

20pages,7 figures,63 references

References in corpus (2)

• The $16$th Hilbert problem on algebraic limit cycles
• Jacobian Conjecture via Differential Galois Theory