Real and p-adic Picard–Vessiot fields

Crespo, Teresa; Hajto, Zbigniew; Put, Marius van der

doi:10.1007/s00208-015-1272-2

Cited by 15 publications

(20 citation statements)

References 12 publications

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“…In [9], T. Crespo, Z. Hajto and M. van der Put show that the condition that the field of constants is algebraically closed may be relaxed. They consider formally real and formally padic fields, whose definition is given in Section 2.…”

Section: Introductionmentioning

confidence: 99%

“…p-adically closed) field of constants are instances of differential fields K such that the field of constants C K is existentially closed in K. Under this hypothesis, the existence of a Picard-Vessiot extension is proved in [12,Theorem 2.2]. In [15] the results in [9] are generalised to the case when C K is existentially closed in K, large and bounded.…”

Section: Introductionmentioning

confidence: 99%

“…In [9], Crespo, Hajto and van der Put obtained the following theorem on existence and uniqueness of Picard-Vessiot fields for formally real ordinary differential fields with real closed field of constants and for formally p-adic ordinary differential fields with p-adically closed field of constants.…”

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Real Liouvillian Extensions of Partial Differential Fields

Crespo¹,

Hajto²,

Mohseni³

2021

SIGMA

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In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally p-adic differential fields with a p-adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally p-adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities of further development of algebraic methods in real dynamical systems.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Real Liouvillian Extensions of Partial Differential Fields

Crespo¹,

Hajto²,

Mohseni³

2021

SIGMA

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“…For a homogeneous linear differential equation defined over a formally real differential field K with a real closed field of constants C K , T. Crespo, Z. Hajto and M. van der Put established in [4] the existence and unicity up to K-differential isomorphism of a formally real Picard-Vessiot extension, endowed with an ordering extending the one in K. In [2] T. Crespo and Z. Hajto characterised real Liouville extensions of ordinary differential fields in terms of real Picard-Vessiot theory answering an earlier question of A. Khovanskii, namely "Is it true that a necessary and sufficient condition for solvability of a real differential equation by real Liouville functions follows from real Picard-Vessiot theory?…”

Section: Introductionmentioning

confidence: 99%

Real Liouvillian Extensions of Partial Differential Fields

Crespo,

Hajto,

Mohseni

2021

Preprint

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In this paper, we consider real partial differential fields with a real closed field of constants. For an integrable differential system defined over such a field, we prove that there exists a real Picard-Vessiot extension. Moreover, two real Picard-Vessiot extensions for the same system with orderings restricting to the same ordering on the base field are isomorphic as differential extensions. We obtain a Galois fundamental theorem in this setting. We characterise real Liouvillian extensions of real partial differential fields by means of split solvable linear algebraic groups. We present some examples of real dynamical systems showing the value of developing Picard-Vessiot theory for real partial differential fields.

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“…a formally p-adic field) with real closed (resp. 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.…”

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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Real and p-adic Picard–Vessiot fields

Cited by 15 publications

References 12 publications

Real Liouvillian Extensions of Partial Differential Fields

Real Liouvillian Extensions of Partial Differential Fields

Real Liouvillian Extensions of Partial Differential Fields

Galois Correspondence Theorem for Picard-Vessiot Extensions

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