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.

arXiv comment20pages,7 figures,63 references

References in corpus (7)

• The $16$th Hilbert problem on algebraic limit cycles
• On polynomial submersions of degree $4$ and the real Jacobian conjecture in ${\mathbb{R}}^2$
• Some Remarks on the Jacobian Conjecture and Dru{ż}kowski mappings
• Some remarks on the Jacobian conjecture and polynomial endomorphisms
• The Jacobian Conjecture fails for pseudo-planes
• Centralizers and normalizers of local analytic and formal vector fields
• Jacobian Conjecture via Differential Galois Theory