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Germs of arcs on singular algebraic varieties and motivic integration. (English) Zbl 0928.14004
In this paper the authors prove a motivic analogue of the rationality of the Poincaré series associated to the $$p$$-adic points on algebraic varieties. They consider the Grothendieck ring on algebraic varieties over a field of characteristic 0: The class of a variety $$S$$ is the sum of the classes of $$S'$$ and $$S''$$ if $$S'$$ is closed in $$S$$ and $$S''$$ is isomorphic to $$S-S'$$. The multiplication is the product of varieties and the ring they consider is a certain subring of the localization of the above Grothendieck ring where multiplication by the affine line is made invertible. The Poincaré series is viewed as an element of the formal power series ring in one variable over the above ring. The main techniques used in the proof are quantifier elimination for semi-algebraic sets of power series in characteristic 0 and M. Kontsevich’s idea of motivic integration.

##### MSC:
 14E18 Arcs and motivic integration 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14B05 Singularities in algebraic geometry 14P10 Semialgebraic sets and related spaces 03C10 Quantifier elimination, model completeness, and related topics 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 14F42 Motivic cohomology; motivic homotopy theory
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