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We lay the groundwork in this first installment of a series of papers aimed at developing a theory of Hrushovski–Kazhdan style motivic integration for certain types of nonarchimedean
-minimal fields, namely power-bounded
-convex valued fields, and closely related structures. The main result of the present paper is a canonical homomorphism between the Grothendieck semirings of certain categories of definable sets that are associated with the
-sort and the
-sort of the language
. Many aspects of this homomorphism can be described explicitly. Since these categories do not carry volume forms, the formal groupification of the said homomorphism is understood as a universal additive invariant or a generalized Euler characteristic. It admits not just one, but two specializations to
. The overall structure of the construction is modeled on that of the original Hrushovski–Kazhdan construction.
We prove that if an equicharacteristic valued field has a ℤ-group as its value group and admits quantifier elimination in the main sort of the prototypical Denef-Pas style language then it is henselian. In fact the proof of this suggests that a reasonable class of Denef-Pas style languages is natural with respect to henselianity.
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