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

Surjectivity of linear operators and semialgebraic global diffeomorphisms