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

TL;DR

This paper explores the relationship between Lipschitz geometry and arc spaces of real algebraic sets by developing a real motivic measure and integral, leading to new inverse mapping theorems and classification tools for singularities.

Contribution

It introduces a real motivic measure and integral, extending motivic integration to real algebraic sets, and applies these tools to Lipschitz geometry and singularity classification.

Findings

Constructed a motivic measure on real analytic arcs.

Established a change of variables formula for real motivic integrals.

Characterized arc-analytic homeomorphisms via motivic measure.

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.