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

Abstract

Let Y : R n → 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 codim(S Y ) ≥ 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.