Injectivity of non-singular planar maps with disconnecting curves in the eigenvalues space - Sabatini, Marco (Topological Methods in Nonlinear Analysis, 2023)
Fessler and Gutierrez \cite{Fe}, \cite{Gu} proved that if a non-singular planarmap has Jacobian matrix without eigenvalues in $(0,+\infty)$, then it is injective.We prove that the same holds replacing $(0,+\infty)$ with any unbounded curve disconnecting the upper (lower) complex half-plane. Additionally we prove thata Jacobian map $(P,Q)$ is injective if $\partial P/\partial x + \partial Q/\partial y$ is not a surjective function.