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Motivic Vitushkin invariants

Part of: Local theory Model theory

Published online by Cambridge University Press:  20 August 2025

G. Comte  and
I. Halupczok
G. Comte
Affiliation:
Univ. Savoie Mont Blanc, CNRS, LAMA, Chambéry, 73000, France georges.comte@univ-smb.fr. http://georgescomte.perso.math.cnrs.fr/
I. Halupczok
Affiliation:
Heinrich Heine University Düsseldorf, Faculty of Mathematics and Natural Sciences, Düsseldorf, 40225, Germany math@karimmi.de. https://immi.karimmi.de/
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Abstract

Using motivic integration theory and the notion of riso-triviality, we introduce two new objects in the framework of definable nonarchimedean geometry: a convenient partial preorder $\preccurlyeq$ on the set of constructible motivic functions, and an invariant $V_0$, nonarchimedean substitute for the number of connected components. We then give several applications based on $\preccurlyeq$ and $V_0$: we obtain the existence of nonarchimedean substitutes of real measure geometric invariants $V_i$, called the Vitushkin variations, and we establish the nonarchimedean counterpart of a real inequality involving $\preccurlyeq$, the metric entropy and our invariants $V_i$. We also prove the nonarchimedean Cauchy–Crofton formula for definable sets of dimension $d$, relating $V_0$ (and $V_d$) and the motivic measure in dimension $d$.

Keywords

metric entropyvitushkin’s invariantsmotivic integration

MSC classification

Primary: 14B05: Singularities 14B07: Deformations of singularities
Secondary: 03C60: Model-theoretic algebra 03C98: Applications of model theory

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 5 , May 2025 , pp. 959 - 992
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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