Integration in power-bounded $T$-convex valued fields

TL;DR

This paper develops a motivic integration theory for power-bounded T-convex valued fields, introducing volume forms and Jacobian-respecting isomorphisms to extend the Grothendieck ring framework.

Contribution

It introduces volume forms valued in the value group and RV-sort, establishing Jacobian-respecting isomorphisms as motivic integrals in T-convex valued fields.

Findings

Established isomorphisms respecting change of variables

Introduced volume forms in the Grothendieck ring setting

Enhanced the framework for applications involving bounded support functions

Abstract

This is the second installment of a series of papers aimed at developing a theory of Hrushovski-Kazhdan style motivic integration for certain types of nonarchimedean $o$-minimal fields, namely power-bounded $T$-convex valued fields, and closely related structures. The main result in the first installment is a canonical isomorphism between the Grothendieck rings of certain categories of definable sets, which is understood as a universal additive invariant or a generalized Euler characteristic because the categories do not carry volume forms. Here we introduce two types of volume forms into each of the relevant categories, one takes values in the value group and the other in the finer RV-sort. The resulting isomorphisms respect Jacobian transformations --- that is, the change of variables formula holds --- and hence are regarded as motivic integrals. As in the classical theory of integration, one is often led to consider locally constant functions with bounded support in various situations. For the space of such functions, the construction may be fine-tuned so as to become more amenable to applications. The modifications are nevertheless substantial and constitute the bulk of the technical work.