Quantifier elimination in valued Ore modules

Abstract

Abstract. We consider valued fields with a distinguished isometry or contractive derivation as valued modules over the Ore ring of difference operators. Under certain assumptions on the residue field, we prove quantifier elimination first in the pure module language, then in that language augmented with a chain of additive subgroups, and finally in a two-sorted language with a valuation map. We apply quantifier elimination to prove that these structures do not have the independence property.

Separably Closed Fields and Contractive Ore Modules

“…[9]), in particular for the above two examples. In [2], we investigated the additive theory of valued fields but with a distinguished isometry (at the opposite of the Frobenius map) and we could obtain results similar to Rohwer's, even at the level of quantifier elimination for such models as K = F p ((T )) with the isometry σ( a i T i ) = a p i T i . In contrast with Rohwer, our starting point does not address directly the structure of some specific classes of definable sets, but is in the spirit of classical elimination of quantifiers algorithms in the theory of modules.…”

“…Under these assumptions, A 0 satisfies the generalized right division algorithm: for any q 1 (t), q 2 (t) ∈ A 0 with deg(q 1 ) ≥ deg(q 2 ), there exist a ∈ D − {0}, d ∈ N and c, r in A 0 with deg(r) < deg(q 2 ) such that q 1 .a d = q 2 .c + r (see e.g. [2], Lemma 2.2).…”

“…(1) the L A -theory of all right A-modules (2) ∀x (x = i∈n λ i (x) · tc i ) (3) ∀x∀(x i ) i∈n (x = i∈n x i · tc i → i∈n x i = λ i (x)).…”

“…Finally let us mention that new results have been obtained by G. Onay on the model theory of valued modules ( [15]), both in the isometric case and the contractive case (the Frobenius map case). We mostly use the notation of [2], with some slight modifications.…”

Bézout domains and lattice-valued modules

“…In section 3, for A a valuation domain, we revisit a quantifier elimination result in the class Mod A , adding to the module language unary predicates for certain pp definable submodules, following the approach of Bélair-Point [2,Proposition 4.1]. This result was essentially known but we need the additional property that for A of the form B M , where B M is the localization of a Bézout domain B at a maximal ideal, to any pp L B M -formula one can associate a constructible subset of MSpec(B) over which the elimination is uniform.…”