Integration in Hensel minimal 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.