Algebraic Groups and Differential Galois Theory

Crespo, Teresa; Hajto, Zbigniew

doi:10.1090/gsm/122

Cited by 63 publications

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“…If the constant c were algebraic over K, it would be algebraic over C K (see e.g. [4], proof of Prop. 3.5).…”

Section: Resultsmentioning

confidence: 99%

“…It was made more accessible by the book of I. Kaplansky [7]. We also refer the reader to [4], [9] and [11] for the results of Picard-Vessiot theory used in this paper.…”

Section: L(y ) := Ymentioning

confidence: 99%

“…It is worth considering whether the condition C K algebraically closed can be weakened. In particular, the case of real fields is interesting due to the application of Picard-Vessiot theory to the integrability of hamiltonian systems (see [10] or [4]). …”

Section: L(y ) := Ymentioning

confidence: 99%

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Picard-Vessiot extensions for real fields

Sowa

2010

Proc. Amer. Math. Soc.

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“…If the constant c were algebraic over K, it would be algebraic over C K (see e.g. [4], proof of Prop. 3.5).…”

Section: Resultsmentioning

confidence: 99%

“…It was made more accessible by the book of I. Kaplansky [7]. We also refer the reader to [4], [9] and [11] for the results of Picard-Vessiot theory used in this paper.…”

Section: L(y ) := Ymentioning

confidence: 99%

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Picard-Vessiot extensions for real fields

Sowa

2010

Proc. Amer. Math. Soc.

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“…Recently, new mathematical approaches based on differential-algebraic [27][28][29][30][31] and differential geometric methods and techniques, were applied in [8,32,33] for studying the Lax integrability of nonlinear differential equations of the Korteweg-de Vries and Riemann type. In particular, many analytical studies [32,[34][35][36][37][38][39][40] …”

Section: Setting the Problem Proposition 3 The Lax Representation Formentioning

confidence: 99%

Hidden Symmetries of Lax Integrable Nonlinear Systems

Blackmore¹,

Prykarpatsky²,

Golenia³

et al. 2013

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Recently devised new symplectic and differential-algebraic approaches to studying hidden symmetry properties of nonlinear dynamical systems on functional manifolds and their relationships to Lax integrability are reviewed. A new symplectic approach to constructing nonlinear Lax integrable dynamical systems by means of Lie-algebraic tools and based upon the Marsden-Weinstein reduction method on canonically symplectic manifolds with group symmetry, is described. Its natural relationship with the well-known Adler-Kostant-Souriau-Berezin-Kirillov method and the associated R-matrix method [1,2] is analyzed in detail. A new modified differential-algebraic approach to analyzing the Lax integrability of generalized Riemann and Ostrovsky-Vakhnenko type hydrodynamic equations is suggested and the corresponding Lax representations are constructed in exact form. The related bi-Hamiltonian integrability and compatible Poissonian structures of these generalized Riemann type hierarchies are discussed by means of the symplectic, gradientholonomic and geometric methods.

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“…For a homogeneous linear differential equation L(Y ) = 0 defined over a differential field K with field of constants C, a Picard-Vessiot extension is a differential field L, differentially generated over K by a fundamental system of solutions of L(Y ) = 0 and with constant field equal to C. A classical result states that the Picard-Vessiot extension exists and is unique up to K -differential isomorphism in the case C algebraically closed (see Kolchin 1948, Crespo and Hajto 2011or Put and Singer 2003. Recently, an existence and uniqueness result for Picard-Vessiot extensions has been established in the case when the differential field K is a formally real field (resp.…”

Section: Introductionmentioning

confidence: 99%

Galois Correspondence Theorem for Picard-Vessiot Extensions

Crespo

Hajto

Adamus³

2015

Arnold Math J.

Self Cite

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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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Algebraic Groups and Differential Galois Theory

Cited by 63 publications

References 0 publications

Picard-Vessiot extensions for real fields

Picard-Vessiot extensions for real fields

Hidden Symmetries of Lax Integrable Nonlinear Systems

Galois Correspondence Theorem for Picard-Vessiot Extensions

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