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

• Campesato Jean-Baptiste
• Fukui Toshizumi
• Kurdyka Krzysztof
• Parusiński Adam

In this paper we investigate the connections between the Lipschitz geometry of real algebraic varieties and the properties of their arc spaces. For this purpose, we first 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 variable formula not only for the birational but also for generically one-to-one Nash maps. A first consequence of this real motivic integration theory is an inverse mapping theorem which holds for continuous rational maps and, more generally, to generically arc-analytic maps. One may note that the latter maps appeared recently in the classification of singularities of real analytic function germs. As an application, we give a condition for a homeomorphism germ between real algebraic varieties to be bi-Lipschitz for the inner metric in terms of the motivic measure.