Quantifier elimination for valued fields equipped with an automorphism

Durhan, Salih; Onay, Gönenç

doi:10.1007/s00029-015-0183-0

Cited by 5 publications

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“…Definition 2.3.5 for precise definitions). Finally Durhan and Onay [DO15] proved that these results hold without any hypothesis on the automorphism. Our second result focuses on the multiplicative setting where we prove an absolute elimination of imaginaries result for the respective model-companions: Theorem B (Theorem 6.2.1).…”

Section: Introductionmentioning

confidence: 89%

Un principe d'Ax-Kochen-Ershov imaginaire

Hils¹,

Rideau-Kikuchi²

2021

Preprint

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We study interpretable sets in henselian and σ-henselian valued fields with value group elementarily equivalent to Q or Z. Our first result is an Ax-Kochen-Ershov type principle for weak elimination of imaginaries in finitely ramified characteristic zero henselian fieldsrelative to value group imaginaries and residual linear imaginaries. We extend this result to the valued difference context and show, in particular, that existentially closed equicharacteristic zero multiplicative difference valued fields eliminate imaginaries in the geometric sorts.On the way, we establish some auxiliary results on separated pairs of characteristic zero henselian fields and on imaginaries in linear structures which are also of independent interest.

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Section: Introductionmentioning

confidence: 89%

Un principe d'Ax-Kochen-Ershov imaginaire

Hils¹,

Rideau-Kikuchi²

2021

Preprint

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“…We say that f ϕ is linear when f is. When ϕ is not of finite order in End(K), the ring R := K[t; ϕ] is isomorphic to ring of linear difference maps equipped with the composition and the addition (see [2] and [3] for more details).…”

Section: Introductionmentioning

confidence: 99%

“…0 is regular for all r, and any x is regular for any monomial and for the scalar 0 R . Regularity is more generally investigated in[3]. Let (K ⊆ M, v), be an extension of characteristic p > 0 valued fields.…”

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Valued Modules Over Skew Polynomial Rings I

Onay¹

2017

J. symb. log.

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Following our first article, we continue to investigate ultrametric modules over a ring of twisted polynomials of the form [K; ϕ], where ϕ is a ring endomorphism of the field K. The main motivation comes from the the theory of valued difference fields (including characteristic p > 0 valued fields equipped with the Frobenius endomorphism). We introduce the class of modules, that we call, affinely maximal and residually divisible and we prove (relative -) quantifier elimination results. Ax-Kochen & Erhov type theorems follow. As an application, we axiomatize, as a valued module, the ultraproduct of algebraically closed valued fields (F p n (t) alg ) n∈N , of fixed characteristic p > 0, each equipped with the morphism x → x p n and with the t-adic valuation.MSC : Primary 03C60; Secondary 03C10, 14G17, 12J20, 16W80, 13J15During this section, we let the pair (K, ϕ) range over difference fields, that is K is a field together with an ring endomorphism ϕ and we let R := K[t; ϕ].Definition 2.1 (K-chains). For a ∈ K, let ·a denotes a unary function symbol (acting on the right). A K-chain, is a structure of the form (∆, <, ∞, (·a) a∈K ) where (∆, <, ∞) is a linear order with a top element ∞ and such that for all non zero a, b ∈ K and γ, δ ∈ ∆ \ {∞}, we have:Remark 2.2. If a ∈ K, and γ is not infinity, then γ · a = ∞ implies a = 0.Notation. In the rest of this paper, whenever (M, v) is a valued structure, a valued field, valued module, valued abelian group etc., for a subset A ⊆ M we denote by vA, the set {v(a) | a ∈ A}.Example 2.3. If (K, v) is a valued field then letting, for λ ∈ K, v(λ) · a := v(λa) = v(λ) + v(a) makes the ordered set vK a K-chain.Conversely we have the following.Proposition 2.4. If ∆ is an infinite K-chain thenis a valuation ring of K.Proof. Same proof as the one given in [8], Proposition 10.Notation. When the K-chain ∆ is clear from the context, we denote by v K the valuation on K given by O ∆ .Theorem 2.5. Set the language L := {<, ·a (a ∈ K), c, ∞}. Then theory of dense K-chains such that ∆ \ {∞} is without end points, together with the axiom ∞ = c is complete and eliminates quantifiers in the language L. Proof. Follows immediately by Théorème 6, and Théorème 13 in [8].Notation. For r ∈ R, let ·r be a unary function symbol. We set L V := {<, ·r r∈R , ∞}, that we call the language of R-chains.Recall that any r ∈ R can be uniquely written as i t i a i , with a i = 0. We call a term t i a i in this expression a monomial of r.A potential jump of an R-chain ∆, is a potential jump for some non zero r. We denote by Jump ∆ (R) the set of all potential jumps in ∆.Remark 2.13.(1) Jump ∆ (R) is an L V -substructure of ∆.(2) Potential jumps of in ∆, can also be defined as the set of ∞ = γ ∈ ∆ such that γ · m 1 = γ · m 2 for all monomials m 1 , m 2 satisfying 0 deg(m 1 ) < deg(m 2 ).(3) Let dcl ∆ denotes the definable closure operator in an R-chain ∆. Then Jump ∆ (R) ⊆ dcl ∆ (∅). Corollary 2.14. Let ∆, ∆ ′ be R-chains such that Jump ∆ (R) ≡ Jump ∆ ′ (R) as L V -structures. Then Jump ∆ (R) and Jump ∆ ′ (R) are isom...

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“…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 [9,11,15,16,24,25]) and, on the other hand, D-structures (a generalization of both difference and differential structures), first for derivations and certain isometries in [31] but also for greater classes of isometries in [4,8,32], and then for automorphisms that might not be isometries [3,17,19,20,27]. 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.…”

Section: Introductionmentioning

confidence: 99%

Some Properties of Analytic Difference Valued Fields

Rideau¹

2015

J. Inst. Math. Jussieu

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We prove field quantifier elimination for valued fields endowed with both an analytic structure and an automorphism that are σ-Henselian. From this result we can deduce various Ax-Kochen-Ersov type results with respect to completeness and the NIP property. The main example we are interested in is the field of Witt vectors on the algebraic closure of F p endowed with its natural analytic structure and the lifting of the Frobenius. It turns out we can give a (reasonable) axiomatization of its first order theory and that this theory is NIP.

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Quantifier elimination for valued fields equipped with an automorphism

Cited by 5 publications

References 12 publications

Un principe d'Ax-Kochen-Ershov imaginaire

Un principe d'Ax-Kochen-Ershov imaginaire

Valued Modules Over Skew Polynomial Rings I

Some Properties of Analytic Difference Valued Fields

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