Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets

Abstract

We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs. We use this measure to define a real motivic integral which admits a change of variables formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for continuous rational maps and, more generally, for generically arc-analytic maps. These maps appeared recently in the classification of singularities of real analytic function germs. Finally, as an application, we characterize in terms of the motivic measure, germs of arc-analytic homeomorphism between real algebraic varieties which are bi-Lipschitz for the inner metric.

References in corpus (5)

• Germs of arcs on singular algebraic varieties and motivic integration
• The thick-thin decomposition and the bilipschitz classification of normal surface singularities
• The Grothendieck ring of varieties is not a domain
• Curve-rational functions
• An inverse mapping theorem for blow-Nash maps on singular spaces