Additive invariants in o-minimal valued fields

TL;DR

This paper develops a motivic integration theory for polynomial-bounded T-convex valued fields within o-minimal structures, establishing homomorphisms between Grothendieck semirings and constructing rational topological zeta functions.

Contribution

It extends motivic integration to non-archimedean o-minimal fields, providing explicit descriptions of generalized Euler characteristics and rationality of zeta functions.

Findings

Established canonical homomorphisms between Grothendieck semirings.

Explicitly described groupifications as generalized Euler characteristics.

Proved rationality of topological zeta functions in this setting.

Abstract

We develop a theory of Hrushovski-Kazhdan style motivic integration for certain type of non-archimedean o-minimal fields, namely polynomial-bounded T-convex valued fields. The structure of valued fields is expressed through a two-sorted first-order language L_TRV. We establish canonical homomorphisms between the Grothendieck semirings of various categories of definable sets that are associated with the VF-sort and the RV-sort of L_TRV. The groupifications of some of these homomorphisms may be described explicitly and are understood as generalized Euler characteristics. In the end, following the Hrushovski-Loeser method, we construct topological zeta functions associated with (germs of) definable continuous functions in an arbitrary polynomial-bounded o-minimal field and show that they are rational. The overall construction is closely modeled on that of the original Hrushovski-Kazhdan construction, as reproduced in the series of papers by the present author.