Special transformations in algebraically closed valued fields

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.Comment: This is the published version of a part of the notes on the Hrushovski-Kazhdan integration theory. To appear in the Annals of Pure and Applied Logi

Quantifier elimination for the reals with a predicate for the powers of two

In 1985, van den Dries showed that the theory of the reals with a predicate for the integer powers of two admits quantifier elimination in an expanded language, and is hence decidable. He gave a model-theoretic argument, which provides no apparent bounds on the complexity of a decision procedure. We provide a syntactic argument that yields a procedure that is primitive recursive, although not elementary. In particular, we show that it is possible to eliminate a single block of existential quantifiers in time $2^0_{O(n)}$, where $n$ is the length of the input formula and $2_k^x$ denotes $k$-fold iterated exponentiation

Integration in algebraically closed valued fields with sections

We construct Hrushovski-Kazhdan style motivic integration in certain expansions of ACVF. Such an expansion is typically obtained by adding a full section or a cross-section from the RV-sort into the VF-sort and some (arbitrary) extra structure in the RV-sort. The construction of integration, that is, the inverse of the lifting map L, is rather straightforward. What is a bit surprising is that the kernel of L is still generated by one element, exactly as in the case of integration in ACVF. The overall construction is more or less parallel to the original Hrushovski-Kazhdan construction. As an application, we show uniform rationality of Igusa zeta functions for non-archimedean local fields with unbounded ramification degrees.Comment: Minor revision in the last sectio