On a weak Jelonek's real Jacobian Conjecture in ${\mathbb{R}}^n$

Let $Y:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be a polynomial local diffeomorphism and let $S_Y$ denote the set of not proper points of $Y$. The Jelonek’s real Jacobian Conjecture states that if $\text{codim}(S_Y)\ge 2$, then $Y$ is bijective. We prove a weak version of such conjecture establishing the sufficiency of a necessary condition for bijectivity. Furthermore, we generalize our result on bijectivity to semialgebraic local diffeomorphisms.