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Germs of arcs on singular algebraic varieties and motivic integration

J. Denef·F. Loeser
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Abstract

We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of the rationality of the Poincare series associated to p-adic points on a p-adic variety. The main tools which are used are semi-algebraic geometry in spaces of power series and motivic integration (a notion introduced by M. Kontsevich). In particular we develop the theory of motivic integration for semi-algebraic sets of formal arcs on singular algebraic varieties, we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of Oesterle.

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J. Denef, F. Loeser. 1998-03-11. Germs of arcs on singular algebraic varieties and motivic integration. https://doi.org/10.1007/s002220050284

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