A semialgebraic map f:X→Y between two real algebraic sets is called
blow-Nash if it can be made Nash (i.e. semialgebraic and real analytic) by
composing with finitely many blowings-up with non-singular centers. We prove
that if a blow-Nash self-homeomorphism f:X→X satisfies a lower
bound of the Jacobian determinant condition then f−1 is also blow-Nash and
satisfies the same condition. The proof relies on motivic integration arguments
and on the virtual Poincar\'e polynomial of McCrory-Parusi\'nski and Fichou. In
particular, we need to generalize Denef-Loeser change of variables key lemma to
maps that are generically one-to-one and not merely birational