Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Johannes Nicaise

Abstract

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.

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

Abstract

Let be a divisible subgroup of . Our central result states that, at the level of Grothendieck groups, the classification of -rational polyhedra in up to affine transformations in is equivalent to the classification up to affine transformations in . 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.
Paper Structure (15 sections, 17 theorems, 74 equations)

This paper contains 15 sections, 17 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Grothendieck rings of polyhedra
  3. $\Gamma$-rational polyhedra
  4. Graded Grothendieck rings of $\Gamma$-rational polyhedra
  5. Rational vs. integral isomorphisms
  6. Euler characteristics on $\mathbf{K}^{\dim}_{\mathbb{Q}}(\Gamma)$.
  7. A Brianchon-Gram decomposition theorem for polyhedra
  8. Generators for $\mathbf{K}^{\dim}(\Gamma)$
  9. Explicit description of $\mathbf{K}^{\dim}(\Gamma)$.
  10. A question by Hrushovski and Kazhdan
  11. Application: the kernel of the motivic volume
  12. The Grothendieck rings of varieties
  13. The Grothendieck ring of semi-algebraic sets
  14. The isomorphism of Hrushovski and Kazhdan
  15. The motivic volume

Key Result

Theorem 3.2.1

For every polytope $P$ in $\mathbb{R}^n$, we have where the sum is taken over all the faces of $P$.

Theorems & Definitions (43)

  • Definition 2.1.1
  • Example 2.1.2
  • Example 2.1.3
  • Example 2.2.1
  • Example 2.2.3
  • Theorem 3.2.1: Brianchon-Gram decomposition theorem for polytopes
  • Proposition 3.2.2
  • proof
  • Corollary 3.2.3
  • proof
  • ...and 33 more