Quelques remarques concernant la théorie des corps ordonnés différentiellement clos

Michaux, Christian; Rivière, Cédric

doi:10.36045/bbms/1126195339

Cited by 12 publications

(9 citation statements)

References 7 publications

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“…Proof As previously noted the proof is a generalization of the proof of Lemma 3.2 appearing in [5] (in this particular case, the topology is the order topology and the role of the largeness hypothesis is played by the intermediate value property for real closed fields).…”

Section: Lemma 35mentioning

confidence: 86%

“…We first recall a definition introduced in [5]: Letā = (a 1 , . .…”

Section: Large Differential Fields and Conditionmentioning

confidence: 99%

“…In [5] the authors use the following lemma to give a less constrained and first-order version of Condition 1 in the particular case of ordered differential fields.…”

Section: Large Differential Fields and Conditionmentioning

confidence: 99%

“…The result stated in Lemma 3.2 can also be obtained in other examples of theories of topological fields. For this, we have to remark that the main tool used in the proof of Lemma 3.2 in [5] is the intermediate value property for real closed fields. So this proof can be slightly modified in order to apply to any theory of topological fields T satisfying a topological analogue of this property.…”

Section: Lemma 32mentioning

confidence: 99%

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Geometrical Axiomatization for Model Complete Theories of Differential Topological Fields

Guzy¹,

Rivière²

2006

Notre Dame J. Formal Logic

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In this paper we give a differential lifting principle which provides a general method to geometrically axiomatize the model companion (if it exists) of some theories of differential topological fields. The topological fields we consider here are in fact topological systems in the sense of van den Dries, and the lifting principle we develop is a generalization of the geometric axiomatization of the theory DCF 0 given by Pierce and Pillay. Moreover, it provides a geometric alternative to the axiomatizations obtained by Tressl and Guzy/Point in separate papers where the authors also build general schemes of axioms for some model complete theories of differential fields. We first characterize the existentially closed models of a given theory of differential topological fields and then, under an additional hypothesis of largeness, we show how to modify this characterization to get a general scheme of first-order axioms for the model companion of any large theory of differential topological fields. We conclude with an application of this lifting principle proving that, in existentially closed models of a large theory of differential topological fields, the jet-spaces are dense in their ambient topological space.

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Section: Lemma 35mentioning

confidence: 86%

“…We first recall a definition introduced in [5]: Letā = (a 1 , . .…”

Section: Large Differential Fields and Conditionmentioning

confidence: 99%

“…In [5] the authors use the following lemma to give a less constrained and first-order version of Condition 1 in the particular case of ordered differential fields.…”

Section: Large Differential Fields and Conditionmentioning

confidence: 99%

Section: Lemma 32mentioning

confidence: 99%

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Geometrical Axiomatization for Model Complete Theories of Differential Topological Fields

Guzy¹,

Rivière²

2006

Notre Dame J. Formal Logic

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“…One transforms a quantifier free formula from the valued difference field language to a formula from the valued field language by replacing a term a' (x) by a new variable. Michaux and Riviere observed in[23] that this kind of trick works in ordered differential fields.…”

mentioning

confidence: 99%

Quantifier elimination in valued Ore modules

Bélair¹,

Point²

2010

J. symb. log.

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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.

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Existentially closed ordered difference fields and rings

Point

2010

Mathematical Logic Qtrly

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Key words Existentially closed difference field, ordered field, valued field, lattice-ordered ring. MSC (2010) 03C60, 12H10, 12J10, 12J15, 13J25We describe classes of existentially closed ordered difference fields and rings. We show an Ax-Kochen type result for a class of valued ordered difference fields. Existentially closed real-closed difference fieldsIn the first part of this paper we will consider on one hand totally ordered difference fields (a difference field is a field with a distinguished automorphism σ) and on the other hand preordered difference fields.By a well-known theorem of A. Tarski, the theory RCF of real-closed fields is the model-companion of the theory of totally ordered fields and a direct consequence of results of H. Kikyo and S. Shelah is that the theory of real-closed totally ordered difference fields, RCF σ , does not have a model-companion (see [17]).Note that in a difference field (K, σ) one has automatically a pair of fields, namely (K, Fix(σ)), where Fix(σ) denotes the subfield of elements of K fixed by σ, and if K is real-closed, then so is Fix(σ). W. Baur showed that the theory of all pairs of real-closed fields (K, L) with a predicate for a subfield is undecidable ([1]). However, he also showed that, adding to the language of ordered rings a new function symbol for a valuation v, the theory of the pairs (K, L) such that v is convex, the residue field of L is dense in the residue field of K and each finitedimensional L-vector space of K has a basis a 1 , . . . , a n satisfying v(First, we describe a class of existentially closed totally ordered difference fields (even though it is not an elementary class). We also consider the case of a proper preordering, using former results of A. Prestel and L. van den Dries (see Section 1.3).Then we consider valued totally-ordered fields and we assume on one hand that σ is strictly increasing on the set of elements of strictly positive valuation and on the other hand that in the pair (K, Fix(σ)) the residue field of K and the residue field of Fix(σ) coincide (and so we are trivially in the Baur setting).We proceed as for the case of valued difference fields with an ω-increasing automorphism treated by E. Hrushovski ([6]) and we show an Ax-Kochen-Ersov type result.In the second part, we consider commutative von Neumann regular lattice-ordered rings with a distinguished automorphism σ which fixes the set of the maximal -ideals and we use transfer results due to S. Burris and H. Werner ([4]) in certain Boolean products in order to describe the class of existentially closed such latticeordered rings.In [15], we showed certain undecidability results for Bezout difference rings. One of the consequences was that any commutative lattice-ordered ring with a distinguished automorphism σ with an infinite orbit on the set of its maximal -ideals has an undecidable theory, whenever the subring fixed by σ is an infinite field ([15, Corollary 8.1]). On the positive side, we also showed that the theory of von Neumann regular commutative f -rings with a pseudo...

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Quelques remarques concernant la théorie des corps ordonnés différentiellement clos

Cited by 12 publications

References 7 publications

Geometrical Axiomatization for Model Complete Theories of Differential Topological Fields

Geometrical Axiomatization for Model Complete Theories of Differential Topological Fields

Quantifier elimination in valued Ore modules

Existentially closed ordered difference fields and rings

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