Mathematische Annalen

, Volume 374, Issue 1–2, pp 211–251 | Cite as

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

  • Jean-Baptiste Campesato
  • Toshizumi Fukui
  • Krzysztof Kurdyka
  • Adam ParusińskiEmail author


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.

Mathematics Subject Classification

14P99 (Primary) 26A16 14E18 14B05 



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Aix Marseille University, CNRS, Centrale Marseille, I2MMarseilleFrance
  2. 2.Department of Mathematics, Faculty of ScienceSaitama UniversitySaitamaJapan
  3. 3.University Grenoble Alpes, University Savoie Mont Blanc, CNRS, LAMAChambéryFrance
  4. 4.Université Côte d’Azur, Université Nice Sophia Antipolis, CNRS, LJADNiceFrance

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