Multiplicative valued difference fields

Abstract

The theory of valued difference fields (K, σ, v) depends on how the valuation v interacts with the automorphism σ. Two special cases have already been worked out -the isometric case, where v(σ(x)) = v(x) for all x ∈ K, has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon [2]; and the contractive case, where v(σ(x)) > nv(x) for all n ∈ N and x ∈ K × with v(x) > 0, has been worked out by Salih Azgin [4]. In this paper we deal with a more general version, called the multiplicative case, where v(σ(x)) = ρ · v(x), where ρ (> 0) is interpreted as an element of a real-closed field. We give an axiomatization and prove a relative quantifier elimination theorem for such a theory.If there is a non-zero L ∈ Z[ρ] such that ∀x > 0(L · x = 0), we say ρ is algebraic (over the integers); otherwise we say ρ is transcendental. If ρ is algebraic, there is a minimal (degree) polynomial that it satisfies.Note that the kernel of Φ need not be trivial. For example, if ρ · x = 2x for all x, then ρ − 2 ∈ Ker(Φ). In particular, Ker(Φ) is non-trivial iff ρ is algebraic. We then form the following ring:Definition 2.3. Let M ODDAG be the L ρ,< -theory of non-trivial multiplicative ordered divisible difference abelian groups. This theory is axiomatized by the above axioms along with ∃x(x = 0) and the following additional infinite list of axioms:i.e., all non-zero linear difference operators are surjective. Thus, M ODDAG is an ∀∃-theory. Similarly as above, we denote by M ODDAG ρ the theory M ODDAG where we fix the order type of ρ.We would now like to show that M ODDAG is the model companion of M ODAG. By abuse of terminology, we would refer to any model of M ODAG (respectively M ODDAG) also as M ODAG (respectively M ODDAG).Remark. It might already be clear from the definitions above that for a given ρ, M ODDAG ρ is basically the theory of non-trivial ordered vector spaces over the ordered field Q(ρ) and then quantifier elimination actually follows from well-known results. However, here we are doing things a little differently. Instead of proving the result for a particular ρ, we are proving it uniformly across all ρ using Axiom OM. And even though in the completion the type of ρ is determined and the theory actually reduces to the above well-known theory, nevertheless it makes sense to write down some of the trivial details just to make sure that nothing fishy happens.Lemma 2.4. M ODAG and M ODDAG are co-theories.Proof. We will actually prove something stronger: for a fixed ρ, M ODAG ρ and M ODDAG ρ are co-theories. Any model of M ODDAG ρ is trivially a model of M ODAG ρ . So all we need to show is that we can embed any model G of M ODAG ρ into a model of M ODDAG ρ .If G is trivial, we can embed it into Q with any given ρ. So without loss of generality we may assume, G is non-trivial.Let Z[ρ] + := {L ∈ Z[ρ] : L > 0}. Define an equivalence relation ∼ on G × Z[ρ] + as follows:(g, L) ∼ (g ′ , L ′ ) ⇐⇒ L ′ · g = L · g ′ .We have thus shown that for a fixed ρ, M ODAG ρ and M ODDAG ρ are cotheories. In fact, since (M ODDAG ρ ) ∀ = ...

Valued difference fields and NTP2

“…One may show in the same way that in the multiplicative case from [Pal12] (see Remark 2.17), the valued difference field is NTP 2 , provided RV (with the induced structure) is NTP 2 .…”

“…In the setting of so-called multiplicative valued difference fields, forming a common generalisation of the contractive and the isometric case, Pal established similar Ax-Kochen-Ershov type results (see [Pal12]), even without adding an angular component map and working in the appropriate language for the RV sort, where RV = K × /1 + m.…”

“…Most of the material in the contractive case may be found in [Azg10], but we will also need the context from [Azg07], where an angular component is used instead of a cross-section. See also [Pal12,Gia11], and the unpublished notes [Hru02] of Hrushovski, where most of the ideas in the contractive case already appear.…”

Some Properties of Analytic Difference Valued Fields

“…There have also been attempts at extending those results to valued fields with more structure. The two most notable enrichments that have been studied are, on the one hand, analytic structures as initiated by [14] and studied thereafter by a great number of people (among many others [32,33,22,23,11,10]) and, on the other hand, D-structures (a generalization of both difference and differential structures), first for derivations and certain isometries in [29] but also for greater classes of isometries in [30,8,4] and then for automorphisms that might not be isometries [3,25,18,17,15]. The model theory of valued differential fields is also quite central to the model theoretic study of transseries (see for example [1]) but the techniques and results in this last field seem quite orthogonal to those in other references given above and to our work here.…”

Model companion of ordered theories with an automorphism

“…However, in his thesis [8], the second-named author noted that if one placed more restrictions on the automorphism, then it is sometimes possible for the expanded theory to have a model companion.…”