Definability of types and VC density in differential topological fields

Point, Françoise

doi:10.1007/s00153-017-0607-y

Cited by 3 publications

(2 citation statements)

References 30 publications

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“…The model theoretic account goes via prime models as follows: The prime model K dif f of K embeds in U over K, has no new constants, and the linear differential equation has a fundamental system of solutions in K dif f as the latter is differentially closed. In the more general situations such as when U is a model of CODF , this approach has no chance of working, as there are no prime models (see [14] and also [12] which adapts Singer's argument to other contexts). But as it turns out we are able to combine the relatively hard abstract existence statements from [5] with some relatively soft model theory to obtain the embedded existence statements and this is the content of the current paper.…”

Section: Introductionmentioning

confidence: 99%

Embedded Picard–Vessiot extensions

Brouette

Cousins

Pillay

et al. 2018

Communications in Algebra

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We prove that if T is a theory of large, bounded, fields of characteristic 0 with almost quantifier elimination, and T D is the model companion of T ∪ {"∂ is a derivation"}, then for any model (U , ∂) of T D , differential subfield K of U such that C K |= T , and linear differential equation ∂Y = AY over K, there is a Picard-Vessiot extension L of K for the equation with K ≤ L ≤ U , i.e. L can be embedded in U over K, as a differential field. Moreover such L is unique to isomorphism over K as a differential field. Likewise for the analogue for strongly normal extensions for logarithmic differential equations in the sense of Kolchin.Recent papers such as [2] and [5] have shown that under certain conditions (on the differential field (K, ∂) and its field C K of constants), given a linear differential equation over (K, ∂) we can find a Picard-Vessiot extension (L, ∂) of (K, ∂) for the equation such that C K is existentially closed in L (as a field). Among the motivating examples to which this applies is the case where C K is * 2. A field K is said to be large (or ample) if for any algebraic variety V which is defined over K, is K-irreducible, and has a nonsingular K-rational point, V (K) is Zariski dense in V .Remark 1.2. 1. Large fields were introduced by Pop in [13] where one can find other characterizations.2. The class of large fields is elementary in the language L.

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Section: Introductionmentioning

confidence: 99%

Embedded Picard–Vessiot extensions

Brouette

Cousins

Pillay

et al. 2018

Communications in Algebra

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“…Using that T c admits quantifier elimination, an easy observation is that whenever T c has N IP , then T * c,D has N IP [15]. This implies in particular that the definable subsets in a model of T * c,D have a V C-dimension and in [34], we related the VC-density of definable sets in T c and in T * c,D . (Note that T c dp-small does not imply that T * c,D dp-small, e.g.…”

Section: Introductionmentioning

confidence: 99%

Definable groups in topological differential fields

Point

2017

Preprint

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For certain theories of existentially closed topological differential fields, we show that there is a strong relationship between L ∪ {D}-definable sets and their Lreducts, where L is a relational expansion of the field language and D a symbol for a derivation. This enables us to associate with an L ∪ {D}-definable group in models of such theories, a local L-definable group. As a byproduct, we show that in closed ordered differential fields, one has the descending chain condition on centralisers.

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Topological fields with a generic derivation

Kovacsics

Point

2023

Annals of Pure and Applied Logic

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Definability of types and VC density in differential topological fields

Cited by 3 publications

References 30 publications

Embedded Picard–Vessiot extensions

Embedded Picard–Vessiot extensions

Definable groups in topological differential fields

Topological fields with a generic derivation

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