Picard-Vessiot theory for real fields

Crespo, Teresa; Hajto, Zbigniew; Sowa, Elzbieta

doi:10.1007/s11856-013-0015-x

Cited by 6 publications

(13 citation statements)

References 17 publications

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“…K ⊂ E ⊂ L, then L|E is a real Picard-Vessiot extension and DGal(L|E) is a C-defined closed subgroup of DGal(L|K). As in the ordinary case (see [5] Theorem 3.1, [6] Theorem 4.4), we obtain a Galois correspondence theorem. Theorem 4.…”

Section: Galois Correspondencementioning

confidence: 53%

“…Let K be a real partial differential field with real closed field of constants C, L|K a real Picard-Vessiot extension. As in the ordinary case (see [6]), we shall consider the set DHom K (L, L(i)) of K-differential morphisms from L into L(i) and transfer the group structure from DAut K(i) L(i) to DHom K (L, L(i)) by means of the bijection…”

Section: Galois Correspondencementioning

confidence: 99%

“…In [5] and [6], we proved the existence of a real Picard-Vessiot extension for linear differential equations defined over a real ordinary differential field with a real closed field of constants C, we gave an appropriate definition of its differential Galois group, proved that it has the structure of a C-defined linear algebraic group and established a Galois correspondence theorem in this setting. In [4], we gave a characterization of real Liouville extensions of an ordinary differential field in terms of differential Galois groups, which answers a question raised in [8].…”

Section: Introductionmentioning

confidence: 99%

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Picard-Vessiot Extensions of Real Differential Fields

Crespo

Hajto

2019

SIGMA

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Section: Galois Correspondencementioning

confidence: 53%

Section: Galois Correspondencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Picard-Vessiot Extensions of Real Differential Fields

Crespo

Hajto

2019

SIGMA

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“…The proofs of Propositions 16 and 17 and Corollary 18 in Crespo et al (2013) remain valid in our present setting. We obtain then the following results.…”

Section: For a Picard-vessiot Extension L|k We Shall Consider The Smentioning

confidence: 68%

“…p-adically closed of the same rank than K ) field of constants C (see Crespo et al 2015). In Crespo et al (2013) we presented a Galois correspondence theorem for Picard-Vessiot extensions of formally real differential fields with real closed field of constants. In this paper we establish a Galois correspondence theorem for general Picard-Vessiot extensions, i.e.…”

Section: Introductionmentioning

confidence: 99%

Galois Correspondence Theorem for Picard-Vessiot Extensions

Crespo

Hajto

Adamus³

2015

Arnold Math J.

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For a homogeneous linear differential equation defined over a differential field K , a Picard-Vessiot extension is a differential field extension of K differentially generated by a fundamental system of solutions of the equation and not adding constants. When K has characteristic 0 and the field of constants of K is algebraically closed, it is well known that a Picard-Vessiot extension exists and is unique up to K -differential isomorphism. In this case the differential Galois group is defined as the group of K -differential automorphisms of the Picard-Vessiot extension and a Galois correspondence theorem is settled. Recently, Crespo, Hajto and van der Put have proved the existence and unicity of the Picard-Vessiot extension for formally real (resp. formally p-adic) differential fields with a real closed (resp. p-adically closed) field of constants. This result widens the scope of application of Picard-Vessiot theory beyond the complex field. It is then necessary to give an accessible presentation of Picard-Vessiot theory for arbitrary differential fields of characteristic zero which eases its use in physical or arithmetic problems. In this paper, we give such a presentation avoiding both the notions of differential universal extension and specializations used by Kolchin and the theories of schemes and Hopf algebras used by other authors. More precisely, we give an adequate definition of the differential Galois group as a linear algebraic group and a new proof of the Galois correspondence theorem for a Picard-Vessiot extension of a differential field with non algebraically closed field of constants, which is more elementary than the existing ones.

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