Additive invariants in o-minimal valued fields

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.

References in corpus (4)

• Monodromy and the Lefschetz fixed point formula
• Quantifier elimination and minimality conditions in algebraically closed valued fields
• Special transformations in algebraically closed valued fields
• Integration in algebraically closed valued fields with sections

Cited by in corpus (1)

• Generalized Euler characteristic in power-bounded T-convex valued fields