Integration in Hensel minimal fields

TL;DR

This paper extends motivic integration to a broad class of Hensel minimal fields, establishing isomorphisms of Grothendieck rings and developing effective dimension and differentiation theories.

Contribution

It generalizes motivic integration frameworks to Hensel minimal fields, including new effective theories and applications to various valued fields.

Findings

Isomorphisms of Grothendieck rings in valued field and leading term sorts

Development of effective 1-h-minimal structures and dimension theory

Application to discretely valued, almost real closed, and pseudo-local fields

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.