On differential Galois groups of strongly normal extensions

Brouette, Quentin; Point, Françoise

doi:10.1002/malq.201600098

Cited by 3 publications

(5 citation statements)

References 33 publications

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“…In fact see Theorems 3 and 4 there for the Galois correspondence. This subsumes [2]. Indeed, part (iii) of Theorem 2.3 that Aut(L/K) is G(A), includes, by virtue of Remark 2.4(2), the statement in [2,Theorem 3.15] that Aut(L/K) is interpretable in the field C K .…”

Section: Strongly Normal Extensionsmentioning

confidence: 65%

“…This subsumes [2]. Indeed, part (iii) of Theorem 2.3 that Aut(L/K) is G(A), includes, by virtue of Remark 2.4(2), the statement in [2,Theorem 3.15] that Aut(L/K) is interpretable in the field C K . In fact one actually obtains definability (rather than just interpretability).…”

Section: Strongly Normal Extensionsmentioning

confidence: 65%

“…In fact one actually obtains definability (rather than just interpretability). Similarly, Remark 2.5 includes the statement in [2,Corollary 3.21] with definability in place of interpretability.…”

Section: Strongly Normal Extensionsmentioning

confidence: 93%

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Some Definable Galois Theory and Examples

Sánchez¹,

Pillay²

2017

Bull. symb. log

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We make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the constants are not “closed” in suitable senses. We also improve the definitions and results on generalized strongly normal extensions from [Pillay, “Differential Galois theory I”, Illinois Journal of Mathematics, 42(4), 1998], using this to give a restatement of a conjecture on almost semiabelian δ-groups from [Bertrand and Pillay, “Galois theory, functional Lindemann–Weierstrass, and Manin maps”, Pacific Journal of Mathematics, 281(1), 2016].

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Section: Strongly Normal Extensionsmentioning

confidence: 65%

Section: Strongly Normal Extensionsmentioning

confidence: 65%

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Some Definable Galois Theory and Examples

Sánchez¹,

Pillay²

2017

Bull. symb. log

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“…, α s ∈ Z 0 ). Theorem 1.1 and its improvements [31,33] have been used, for example, in algorithms and effective bounds for differential-algebraic equations [9,10,12,14,26], Galois theory of differential and difference equations [5,16], model theory of differential fields [25,38], control theory [3,13], and for connecting algebraic and analytic approaches to differential-algebraic equations [34,35,36].…”

Section: Overview and Prior Resultsmentioning

confidence: 99%

A primitive element theorem for fields with commuting derivations and automorphisms

Pogudin

2019

Sel. Math. New Ser.

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We establish a Primitive Element Theorem for fields equipped with several commuting operators such that each of the operators is either a derivation or an automorphism. More precisely, we show that for every extension F ⊂ E of such fields of zero characteristic such that • E is generated over F by finitely many elements using the field operations and the operators,• every element of E satisfies a nontrivial equation with coefficient in F involving the field operations and the operators,• the action of the operators on E is irredundant there exists an element a ∈ E such that E is generated over F by a using the field operations and the operators. This result generalizes the Primitive Element Theorems by Kolchin and Cohn in two directions simultaneously: we allow any numbers of derivations and automorphisms and do not impose any restrictions on the base field F .

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“…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. See also Lemma 4.4 of [1] and the paragraph following it which discuss related issues.…”

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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On differential Galois groups of strongly normal extensions

Cited by 3 publications

References 33 publications

Some Definable Galois Theory and Examples

Some Definable Galois Theory and Examples

A primitive element theorem for fields with commuting derivations and automorphisms

Embedded Picard–Vessiot extensions

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