An inverse mapping theorem for blow-Nash maps on singular spaces

Abstract

A semialgebraic map $f:X\to Y$ 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 $f:X\to X$ 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.

References in corpus (1)

• Inverse Function Theorems for Arc-analytic Homeomorphisms

Cited by in corpus (3)

• Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets
• Complete classification of Brieskorn polynomials up to the arc-analytic equivalence
• On the arc-analytic type of some weighted homogeneous polynomials