Grothendieck rings of polytopes and non-archimedean semi-algebraic sets

Abstract

Let $\Gamma$ be a divisible subgroup of $(\mathbb{R},+)$. Our central result states that, at the level of Grothendieck groups, the classification of $\Gamma$-rational polyhedra in $\mathbb{R}^n$ up to affine transformations in $\Gamma^n\rtimes \mathrm{GL}_n(\mathbb{Z})$ is equivalent to the classification up to affine transformations in $\Gamma^n\rtimes \mathrm{GL}_n(\mathbb{Q})$. We prove this by giving an explicit description of these Grothendieck groups. This yields, in particular, a positive answer to the basic case of a question by Hrushovski and Kazhdan; all other cases are still open. As a second application, we give a simple description of the kernel of the motivic volume for non-archimedean semi-algebraic sets, which is a key ingredient of Hrushovski and Kazhdan's theory of motivic integration.