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A p-adic variant of Kontsevich–Zagier integral operation rules and of Hrushovski–Kazhdan style motivic integration

Raf Cluckers and Immanuel Halupczok


We prove that if two semi-algebraic subsets of pn have the same p-adic measure, then this equality can already be deduced using only some basic integral transformation rules. On the one hand, this can be considered as a positive answer to a p-adic analogue of a question asked by Kontsevich–Zagier in the reals (though the question in the reals is much harder). On the other hand, our result can also be considered as stating that over p, universal motivic integration (in the sense of Hrushovski–Kazhdan) coincides with the usual p-adic integration.

Dedicated to Angus Macintyre, source of inspiration

Funding source: European Research Council

Award Identifier / Grant number: 615722 MOTMELSUM

Funding source: KU Leuven

Award Identifier / Grant number: IF C14/17/083

Funding source: Agence Nationale de la Recherche

Award Identifier / Grant number: ANR-11-LABX-0007-01

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: SFB 878

Award Identifier / Grant number: GRK 2240

Award Identifier / Grant number: 426488848

Funding statement: Raf Cluckers was partially supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement No. 615722 MOTMELSUM, by KU Leuven IF C14/17/083, and thanks the Labex CEMPI (ANR-11-LABX-0007-01). Immanuel Halupczok was partially supported by the SFB 878: Groups, Geometry and Actions, by the research training group GRK 2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology, and by the individual research grant No. 426488848, Archimedische und nicht-archimedische Stratifizierungen höherer Ordnung, all three funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation). Part of the work has been done while Immanuel Halupczok was affiliated to the University of Leeds.


We would like to thank J. Denef, F. Loeser and A. Macintyre for interesting discussions on the topics of the paper, and the referees for useful comments and suggestions.


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Received: 2020-03-05
Revised: 2021-03-11
Published Online: 2021-08-14
Published in Print: 2021-10-01

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