Special transformations in algebraically closed valued fields

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 [8]. 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, Γ)-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.

Integration in algebraically closed valued fields

“…In this paper we present the third step. We mention again what has been said in the introduction of [12] that, conceptually, the Hrushovski-Kazhdan construction of motivic integration is completed by the first two steps. Besides the obvious benefit of a deeper understanding, the main reason that we would like to carry out the third step is to facilitate computation in future applications.…”

“…This is understood as a motivic integration, since Fubini's theorem (Theorem 7.14) and the change of variables formula (Theorem 7.15) hold. The first two steps are presented in [12]. In this paper we present the third step.…”

“…The fundamental idea is to construct homomorphisms between various Grothendieck rings associated with the first-order theory ACVF S (0, 0) of algebraically closed valued fields as naturally formulated in the language L RV . This construction has three main steps, which are described in the introduction of [12]. For clarity we shall briefly recall what they are here.…”

“…Throughout this paper we shall use the notation and terminology of [11,12], some of which are recalled in Section 2, where we also review the results in [11,12] that shall be cited later. In Section 3 we prove some structural properties concerning the basic geography of definable sets in ACVF S (0, 0).…”

“…For clarity we shall briefly recall what they are here. Let µVF * and µRV[ * ] be two categories of definable sets with volume forms that are respectively associated with the VF-sort and the RV-sort of L RV (see [12,Section 10]). To construct a canonical homomorphism from the Grothendieck semigroup K + µVF * to the Grothendieck semigroup K + µRV[ * ]/µI sp , where µI sp is a suitable semigroup congruence relation, we proceed as follows:…”