Quantifier elimination and minimality conditions in algebraically closed valued fields

Abstract

A Basarab-Kuhlmann style language L RV is introduced in the Hrushovski-Kazhdan integration theory [10]. The theory ACVF of algebraically closed valued fields formulated in this language admits quantifier elimination, which is not proved in [10] and the reader is referred to a result about a much more complicated language. In this paper, using well-known facts in the theory of valued fields, we give a straightforward proof. We also show that two expansions ACVF † and ACVF ‡ of ACVF, one with a section of the entire RV-sort and the other with a section of the residue field, admit quantifier elimination. Thereafter we show that, in terms of certain minimality conditions, the three theories are distinct geometrically.

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

“…The construction is more or less a variation of that in the proof of [18,Theorem 3.10]. The strategy is to reduce the situation to Theorem 2.11.…”

“…This may be further divided into several cases according to whether a, b are open or closed discs and whether the ends of A are open or closed. In each of these cases, Lemma 3.20 is applied in much the same way as its counterpart is applied in the proof of [18,Lemma 4.26]. It is a tedious exercise and is left to the reader.…”

Bounded integral and motivic Milnor fiber

“…Suppose for contradiction that f (A) is not bounded. For each c ∈ M 0, by [12,Corollary 4.18], acl(c) is a model of ACVF(S) and hence A val(c) ∩ acl(c) is nonempty. Then, by compactness, there is a definable finitary function g : M {0} −→ P(A) such that val(f (g(x))) < − val(x) for every x ∈ M {0}.…”

“…For each γ ∈ Γ, let A γ = {a ∈ A : p(a) > γ}. For each c ∈ M {0}, by [12,Corollary 4.18], acl(c) is a model of ACVF(S) and hence A val(c) ∩ acl(c) is nonempty. By compactness, there is a definable finitary function f : M {0} −→ A such that for each c ∈ M {0} and each a ∈ f (c), p(a) > val(c).…”

Fourier transform of the additive group in algebraically closed valued fields

“…The reader is referred to [19,21,20,23] for notation and terminology. There will be reminders and, inevitably, repetitions as we go along.…”