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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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References

Bernig, A. and Bröcker, L., Lipschitz-Killing invariants, Math. Nachr. 245 (2002), 5–25.Google Scholar
Bradley-Williams, D. and Halupczok, I., Spherically complete models of hensel minimal valued fields, Preprint (2021), arXiv:2109.08619.Google Scholar
Bradley-Williams, D. and Halupczok, I., Riso-stratifications and a tree invariant, Preprint (2022), arXiv:2206.03438.Google Scholar
Bürgisser, P., Kulkarni, A. and Lerario, A., Nonarchimedean integral geometry, Preprint (2022), arXiv:2206.03708.Google Scholar
Cluckers, R., Comte, G. and Loeser, F., Local metric properties and regular stratifications of $p$ -adic definable sets, Comment. Math. Helv. 87 (2012), 963–1009.Google Scholar
Cluckers, R., Halupczok, I. and Rideau-Kikuchi, S., Hensel minimality i, Forum Math. Pi 10 (2022), e11.Google Scholar
Cluckers, R. and Loeser, F., Constructible motivic functions and motivic integration, Invent. Math. 173 (2008), 23–121.Google Scholar
Cluckers, R. and Loeser, F., Motivic integration in all residue field characteristics for Henselian discretely valued fields of characteristic zero, J. Reine Angew. Math. 701 (2015), 1–31.Google Scholar
Comte, G., Équisingularité réelle: nombres de Lelong et images polaires, Ann. Sci. Éc. Norm. Supér. 33 (2000), 757–788.Google Scholar
Comte, G., Deformation of singularities and additive invariants, J. Singul. 13 (2015), 11–41.Google Scholar
Forey, A., Motivic local density, Math. Z. 287 (2017), 361–403.Google Scholar
Forey, A., A motivic local Cauchy-Crofton formula, Manuscripta Math. 166 (2021), 523–533.Google Scholar
Fu, J. H. G., Tubular neighborhoods in Euclidean spaces, Duke Math. J. 52 (1985), 1025–1046.Google Scholar
Golan, J. S., Semirings and their applications (Kluwer Academic, Dordrecht, 1999), updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science (Longman Scientific & Technical, Harlow, 1992); MR 1163371 (93b:16085).Google Scholar
Halupczok, I., Non-archimedean Whitney stratifications, Proc. Lond. Math. Soc. (3) 109 (2014), 1304–1362.Google Scholar
Hrushovski, E. and Kazhdan, D., Integration in valued fields, in Algebraic geometry and number theory, Progress in Mathematics, vol. 253 (Birkhäuser Boston, Boston, MA, 2006), 261–405.CrossRefGoogle Scholar
Ivanov, L. D., Variatsii mnozhestv i funktsi $ \breve{i} $ , edited by A. G. Vituškin (Izdat. “Nauka”, Moscow, 1975).Google Scholar
Kulkarni, A. and Lerario, A., $p$ -adic integral geometry, SIAM J. Appl. Algebra Geom. 5 (2021), 28–59.Google Scholar
M. Raibaut, Analyse et géométrie non-archimédiennes fronts d’ondes & singularités, Mémoire d’habilitation à diriger des recherches (2020).Google Scholar
Takahashi, M., On the bordism categories. III. Functors Hom and $\otimes$ for semimodules, Math. Semin. Notes 10 (1982), 211–236.Google Scholar
Vituškin, A. G., O mnogomernyh variaciyah (Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1955).Google Scholar
Weyl, H., On the volume of tubes, Amer. J. Math. 61 (1939), 461–472.Google Scholar
Yomdin, Y. and Comte, G., Tame geometry with application in smooth analysis, Lecture Notes in Mathematics, vol. 1834 (Springer-Verlag, Berlin, 2004).Google Scholar
Zerner, M., Approximation par des variétés algébriques dans les espaces hilbertiens (Variation sur un thème de Vituškin), Studia Math. 38 (1970), 413–435.Google Scholar