AN INVERSE MAPPING THEOREM FOR BLOW-NASH MAPS ON SINGULAR SPACES

A semialgebraic map $f:X\rightarrow Y$ 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 nonsingular centers.We prove that if a blow-Nash self-homeomorphism $f:X\rightarrow X$ 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 Poincaré polynomial of McCrory–Parusiński 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.