Special transformations in algebraically closed valued fields

TL;DR

This paper simplifies the construction of motivic integration in algebraically closed valued fields by focusing on a syntactically manageable subclass of V-minimal theories, streamlining key steps in the process.

Contribution

It introduces a simplified approach to motivic integration for a specific subclass of algebraically closed valued fields, leveraging syntax to clarify the construction process.

Findings

Constructs a homomorphism between Grothendieck semigroups for ACVF_S(0,0)

Simplifies technical arguments in motivic integration theory

Focuses on a syntactically explicit subclass of V-minimal theories

Abstract

We present two of the three major steps in the construction of motivic integration, that is, a homomorphism between Grothendieck semigroups that are associated with a first-order theory of algebraically closed valued fields, in the fundamental work of Hrushovski and Kazhdan. We limit our attention to a simple major subclass of V-minimal theories of the form ACVF_S(0, 0), that is, the theory of algebraically closed valued fields of pure characteristic $0$ expanded by a (VF, Gamma)-generated substructure S in the language L_RV. The main advantage of this subclass is the presence of syntax. It enables us to simplify the arguments with many different technical details while following the major steps of the Hrushovski-Kazhdan theory.