Grothendieck rings of polytopes and non-archimedean semi-algebraic sets
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914684848832512 |
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| author | Nicaise, Johannes |
| author_facet | Nicaise, Johannes |
| contents | Let $Γ$ be a divisible subgroup of $(\mathbb{R},+)$. Our central result states that, at the level of Grothendieck groups, the classification of $Γ$-rational polyhedra in $\mathbb{R}^n$ up to affine transformations in $Γ^n\rtimes \mathrm{GL}_n(\mathbb{Z})$ is equivalent to the classification up to affine transformations in $Γ^n\rtimes \mathrm{GL}_n(\mathbb{Q})$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11699 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Grothendieck rings of polytopes and non-archimedean semi-algebraic sets Nicaise, Johannes Algebraic Geometry Combinatorics Logic Let $Γ$ be a divisible subgroup of $(\mathbb{R},+)$. Our central result states that, at the level of Grothendieck groups, the classification of $Γ$-rational polyhedra in $\mathbb{R}^n$ up to affine transformations in $Γ^n\rtimes \mathrm{GL}_n(\mathbb{Z})$ is equivalent to the classification up to affine transformations in $Γ^n\rtimes \mathrm{GL}_n(\mathbb{Q})$. 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. |
| title | Grothendieck rings of polytopes and non-archimedean semi-algebraic sets |
| topic | Algebraic Geometry Combinatorics Logic |
| url | https://arxiv.org/abs/2402.11699 |