Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Integration in Hensel minimal fields

Mathias Stout, Floris Vermeulen

TL;DR

This work constructs a universal motivic integration framework for $1$-h-minimal theories of equicharacteristic zero, extending Hrushovski–Kazhdan and Yin to broad valued-field settings and enabling a precise comparison between Grothendieck semirings of ${\mathrm{VF}}$-definable sets and ${\mathrm{RV}}$-definable sets. Central to the approach is the lifting map ${\mathfrak L}$ and an integration map ${\mathcal I}$ that identify ${\mathrm{K}}_+ {\mathrm{VF}}$ with ${\mathrm{K}}_+ {\mathrm{RV}}[*]/I_{\mathrm{sp}}$ (and measured variants with $I_{\mathrm{sp}}^{\mu}$) under the assumption of effectivity, thereby generalizing HK and Yin.tcon. The paper systematically develops dimension theory on ${\mathrm{RV}}$, a robust theory of differentiation and Jacobians in the RV-sort, and a measured extension capturing volume forms and intrinsic RV-Jacobians, with applications to universal motivic invariants and potential connections to Cluckers–Loeser motivic volumes. The results establish a solid foundation for future work on motivic distributions, Fourier–Mellin transforms, and mixed-characteristic generalizations in a tamely behaved ${\mathrm{RV}}$-geometric setting.

Abstract

We develop a framework of motivic integration in the style of Hrushovski--Kazhdan in arbitrary Hensel minimal fields of equicharacteristic zero. Hence our work generalizes that of Hrushovski--Kazhdan and Yin, but applies more broadly to discretely valued fields, almost real closed fields with analytic structure, pseudo-local fields, and coarsenings. In more detail, we obtain isomorphisms of Grothendieck rings of definable sets, with or without volume forms, in the valued field sort and in the leading term sort. Along the way we develop a theory of effective 1-h-minimal structures, where finite definable sets can be lifted from the leading term sort to the valued fields sort. We show that many natural examples of 1-h-minimal structures are effective, and develop dimension theory and a theory of differentiation in $\mathrm{RV}$ for effective structures.

Integration in Hensel minimal fields

TL;DR

This work constructs a universal motivic integration framework for -h-minimal theories of equicharacteristic zero, extending Hrushovski–Kazhdan and Yin to broad valued-field settings and enabling a precise comparison between Grothendieck semirings of -definable sets and -definable sets. Central to the approach is the lifting map and an integration map that identify with (and measured variants with ) under the assumption of effectivity, thereby generalizing HK and Yin.tcon. The paper systematically develops dimension theory on , a robust theory of differentiation and Jacobians in the RV-sort, and a measured extension capturing volume forms and intrinsic RV-Jacobians, with applications to universal motivic invariants and potential connections to Cluckers–Loeser motivic volumes. The results establish a solid foundation for future work on motivic distributions, Fourier–Mellin transforms, and mixed-characteristic generalizations in a tamely behaved -geometric setting.

Abstract

We develop a framework of motivic integration in the style of Hrushovski--Kazhdan in arbitrary Hensel minimal fields of equicharacteristic zero. Hence our work generalizes that of Hrushovski--Kazhdan and Yin, but applies more broadly to discretely valued fields, almost real closed fields with analytic structure, pseudo-local fields, and coarsenings. In more detail, we obtain isomorphisms of Grothendieck rings of definable sets, with or without volume forms, in the valued field sort and in the leading term sort. Along the way we develop a theory of effective 1-h-minimal structures, where finite definable sets can be lifted from the leading term sort to the valued fields sort. We show that many natural examples of 1-h-minimal structures are effective, and develop dimension theory and a theory of differentiation in for effective structures.
Paper Structure (37 sections, 89 theorems, 153 equations)

This paper contains 37 sections, 89 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Further avenues
  4. Outline
  5. Acknowledgements
  6. Preliminaries
  7. Semigroups and semirings
  8. Grothendieck semigroups and semirings
  9. Basic notation in valued fields
  10. Some model-theoretic conventions
  11. Hensel minimality and cell decomposition
  12. Differentiation and the Jacobian property
  13. On $\mathrm{acl}$-dimension
  14. Effective theories
  15. The definition and basic properties
  16. ...and 22 more sections

Key Result

Theorem 1.1.1

Let ${\mathcal{T}}$ be a $1$-h-minimal theory of equicharacteristic zero valued fields. Then there is a natural surjective morphism of semirings where $I_{\mathrm{sp}} = ([1]_1, [\mathrm{RV}^\times_{<1}]_1 + [1]_0)$, such that for every $\emptyset$-definable $R\subset (\mathrm{RV}^\times)^n$ we have $\int([\operatorname{rv}^{-1}(R)]_n) = [R]_n$.

Theorems & Definitions (218)

  • Theorem 1.1.1: Theorem \ref{['thm:non.eff.int']}
  • Theorem 1.1.2: Theorem \ref{['thm:integration.semiring']}
  • Theorem 1.1.3: Theorem \ref{['thm:isom.K.mu']}
  • Example 2.1.1
  • Example 2.1.2
  • Example 2.2.1
  • Example 2.2.2
  • Proposition 2.5.1: Uniform preparation CHR
  • Definition 2.5.2: Twisted boxes
  • Definition 2.5.3: Cell decomposition
  • ...and 208 more