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

Abstract

Abstract We prove that if two semi-algebraic subsets of ℚ p n {\mathbb{Q}_{p}^{n}} 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 ℚ p {\mathbb{Q}_{p}} , universal motivic integration (in the sense of Hrushovski–Kazhdan) coincides with the usual p-adic integration.