Surjectivity of linear operators and semialgebraic global diffeomorphisms

We prove that a $C^{\infty}$ semialgebraic local diffeomorphism of $\mathbb{R}^n$ with non-properness set having codimension greater than or equal to $2$ is a global diffeomorphism if $n-1$ suitable linear partial differential operators are surjective. Then we state a new analytic conjecture for a polynomial local diffeomorphism of $\mathbb{R}^n$. Our conjecture implies a very known conjecture of Z. Jelonek. We further relate the surjectivity of these operators with the fibration concept and state a general global injectivity theorem for semialgebraic mappings which turns out to unify and generalize previous results of the literature.