Some Definable Galois Theory and Examples

Sánchez, Omar León; Pillay, Anand

doi:10.1017/bsl.2017.2

Cited by 14 publications

(8 citation statements)

References 20 publications

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“…We will need to know something about the Galois group associated to an X-strongly normal extension. This is contained in Theorem 2.3 of [11]. But we give a summary.…”

Section: Generalized Picard-vessiot Extensions and Variantsmentioning

confidence: 85%

“…, δ m } of derivations. We first recall a definition from [11] (Definition 3.3) which is itself a slight elaboration on a notion from [17].…”

Section: Generalized Picard-vessiot Extensions and Variantsmentioning

confidence: 99%

“…-Let K be a (small) subfield of U, X some K-definable set, and L a differential field extension of K which is finitely generated over K (as a differential field). L is said to be an X-strongly normal extension of K if 1and 2hold), the type q is isolated (see [11,Lemma 2…”

Section: Generalized Picard-vessiot Extensions and Variantsmentioning

confidence: 99%

“…This, and some variants, is carried out in Section 3, where we also adapt cohomological arguments of Kolchin. In any case our generalized PV theory will be a special case of the "generalized strongly normal theory" from [17] and [11], but still properly include Landesman's theory [10] (and the so-called parameterized Picard-Vessiot theory). So in the case of m > 1 condition (1) will be replaced by "K is algebraically closed and has no proper generalized Picard-Vessiot extensions" (in fact something slightly stronger).…”

Section: Introductionmentioning

confidence: 99%

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Generalized Picard–Vessiot extensions and differential Galois cohomology

Chatzidakis¹,

Pillay²

2020

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques » (http://afst.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://afst. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/ Annales de la faculté des sciences de Toulouse Volume XXVIII, n o 5, 2019 pp. 813-830 Generalized Picard-Vessiot extensions and differential Galois cohomology (*) Zoé Chatzidakis (1) and Anand Pillay (2) ABSTRACT.-In [18] it was proved that if a differential field (K, δ) of characteristic 0 is algebraically closed and closed under Picard-Vessiot extensions then every differential algebraic PHS over K for a linear differential algebraic group G over K has a K-rational point (in fact if and only if). This paper explores whether and if so, how, this can be extended to (a) several commuting derivations, (b) one automorphism. Under a natural notion of "generalized Picard-Vessiot extension" (in the case of several derivations), we give a counterexample. We also have a counterexample in the case of one automorphism. We also formulate and prove some positive statements in the case of several derivations. RÉSUMÉ.-On a montré dans [18] que si un corps différentiel (K, δ) de caractéristique 0 est algébriquement clos et clos par extensions de Picard-Vessiot, alors tout espace principal homogène différentiel algébrique sur K a un point K-rationnel (et réciproquement). Cet article explore s'il est possible, et si oui comment, d'étendre ce résultat au cas de (a) plusieurs dérivations qui commutent, (b) un automorphisme. Pour une notion naturelle d'« extension de Picard-Vessiot généralisée » (dans le cas de plusieurs dérivations) nous donnons un contre-exemple. Nous avons aussi un contre-exemple dans le cas d'un automorphisme. Enfin, nous formulons et démontrons quelques résultats positifs dans le cas de plusieurs dérivations.

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“…We will need to know something about the Galois group associated to an X-strongly normal extension. This is contained in Theorem 2.3 of [11]. But we give a summary.…”

Section: Generalized Picard-vessiot Extensions and Variantsmentioning

confidence: 85%

“…, δ m } of derivations. We first recall a definition from [11] (Definition 3.3) which is itself a slight elaboration on a notion from [17].…”

Section: Generalized Picard-vessiot Extensions and Variantsmentioning

confidence: 99%

Section: Generalized Picard-vessiot Extensions and Variantsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized Picard–Vessiot extensions and differential Galois cohomology

Chatzidakis¹,

Pillay²

2020

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…We will assume familiarity with differential Galois theory, which is the theory of definable automorphism groups in the special case of DCF 0 . The paper [9] has a discussion of differential Galois theory which includes definitions and details around the material below (Picard-Vessiot, strongly normal, extensions of differential fields). Proposition 4.9.…”

Section: The Canonical Base Propertymentioning

confidence: 99%

Relative internality and definable fibrations

Jaoui¹,

Léo²,

Pillay³

2020

Preprint

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We first elaborate on the theory of relative internality in stable theories from [7], focusing on the notion of uniform relative internality (called collapse of the groupoid in [7]), and relating it to orthogonality, triviality of fibrations, the strong canonical base property, differential Galois theory, and GAGA.We prove that DCF 0 does not have the strong canonical base property, correcting a proof in [14]. We also prove that the theory CCM of compact complex manifolds does not have the strong CBP, and initiate a study of the definable Galois theory of projective bundles. In the rest of the paper we study definable fibrations in DCF 0 , where the general fibre is internal to the constants, including differential tangent bundles, and geometric linearizations. We obtain new examples of higher rank types orthogonal to the constants. Contents 1. Introduction 1 2. Preliminaries 3 3. Uniform Relative Internality 5 4. The canonical base property 9 4.1. The strong CBP 9 4.2. DCF 0 10 4.3. CCM 11 4.4. Pairs of algebraically closed fields 14 5. Some effective criteria for uniform internality in DCF 0 15 5.1. Case of the logarithmic derivative 15 5.2. Case of the derivative 17 6. Lifting orthogonality to the constants from a hypersurface 18 6.1. Linearization along an invariant subvariety 18 6.2. Non-uniform invariant hypersurfaces 20 6.3. Application to autonomous differential equations 21 7. The Kolchin tangent bundle and uniform relative internality 22 7.1. A counterexample 23 References 25

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On differential Galois groups of strongly normal extensions

Brouette

Point

2018

Mathematical Logic Qtrly

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We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or p‐valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in sans-serifCODF, we establish a relative Galois correspondence for relatively definable subgroups of the group of differential order automorphisms.

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Some Definable Galois Theory and Examples

Cited by 14 publications

References 20 publications

Generalized Picard–Vessiot extensions and differential Galois cohomology

Generalized Picard–Vessiot extensions and differential Galois cohomology

Relative internality and definable fibrations

On differential Galois groups of strongly normal extensions

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