Picard Vessiot Research Articles

Equivariant vector fields on non-trivial [formula omitted]-torsors and differential Galois theory

We show how to produce SO n -equivariant vector fields on non-trivial SO n -torsors which correspond to quadratic forms non-equivalent to the unit form. For n ⩾ 3 we then give an example of a Picard–Vessiot extension with group SO n which is the function field of a non-trivial SO n -torsor.

Generic rings for Picard–Vessiot extensions and generic differential equations

Let G be an observable subgroup of G L n . We produce an extension of differential commutative rings generic for Picard–Vessiot extensions with group G .

Liouville Extensions of Artinian Simple Module Algebras

In a previous article (Amano and Masuoka, 2005), the author and Masuoka developed a Picard–Vessiot theory for module algebras over a cocommutative pointed smooth Hopf algebra D. By using the notion of Artinian simple (AS)D-module algebras, it generalizes and unifies the standard Picard–Vessiot theories for linear differential and difference equations. The purpose of this article is to define the notion of Liouville extensions of AS D-module algebras and to characterize the corresponding Picard–Vessiot group schemes.

Remarks on a Lie Theorem on the Integrability of Differential Equations in Closed Form

is integrable by quadratures [1] (for details, see also [2, 3]). More precisely, all of its solutions can be found by “algebraic operations” (including inversion of functions) and “quadratures,” that is, computation of integrals of known functions of one variable. This definition has a local nature. The algebra g generates an n-dimensional solvable Lie group G that acts freely on R. Relations (1) imply that the transformations in G take the trajectories of system (2) to trajectories of the same system. In other words, the solvable group G “permutes” the trajectories of the differential equation (2). Lie treated his theorem as an analog of Galois theory for ordinary differential equations. However, there is a more natural analog of Galois theory, namely, Picard–Vessiot theory, which studies extensions of differential fields by solutions of linear differential equations (e.g., see [4]). Nevertheless, the Lie theorem is one of the key results in the problem of closed-form integrability of nonlinear differential equations. It suffices to note that, in the simplest case, this theorem readily implies the well-known Liouville theorem on the integrability of Hamiltonian differential equations with a complete set of independent first integrals in involution [3]. The following assertion is the main result of the present paper.

Picard–Vessiot extensions of artinian simple module algebras

This paper pursues Takeuchi's Hopf algebraic approach [M. Takeuchi, A Hopf algebraic approach to the Picard–Vessiot theory, J. Algebra 122 (1989) 481–509] to the Picard–Vessiot (PV) theory for differential equations, to involve the PV extensions of difference equations. Differential fields and total difference rings in the standard PV theory are unified here by artinian simple (AS) module algebras over a cocommutative, pointed smooth Hopf algebra.

Valuations invariantes pour l'action des groupes de Galois différentiels

Let ( F / K , ∂ ) be a differential field extension with differential Galois group G = Gal ∂ ( F / K ) . For the natural action of G on the Riemann–Zariski variety S ⋆ = S ⋆ ( F / K ) of the field extension F / K , we study the invariant valuations ν ∈ ( S ⋆ ) G when they do exist. We show close relations between these invariant valuations and the elements of F holonomic over K. Next, we study the continuity of the derivation ∂ with respect to these ν-adic topologies. We give a geometric structure property of G-invariant valuation inspired by Zariski. Finally, we give an answer for the existence problem of invariant valuations in the context of Picard–Vessiot extension. To cite this article: G. Duval, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Quasialgebraicity of Picard—Vessiot Fields

We prove that under certain spectral assumptions on the monodromy group, solutions of Fuchsian systems of linear equations on the Riemann sphere admit explicit global bounds on the number of their isolated zeros.

The Picard–Vessiot Antiderivative Closure

F is a differential field of characteristic zero with algebraically closed field of constants C. A Picard–Vessiot antiderivative closure of F is a differential field extension E⊃F which is a union of Picard–Vessiot extensions of F, each obtained by iterated adjunction of antiderivatives, and such that every such Picard–Vessiot extension of F has an isomorphic copy in E. The group G of differential automorphisms of E over F is shown to be prounipotent. When C is the complex numbers and F=C(t) the rational functions in one variable, G is shown to be free prounipotent.

Différentielles non commutatives et théorie de Galois différentielle ou aux différences

We show how the Galois–Picard–Vessiot theory of differential equations and difference equations, and the theory of holonomy groups in differential geometry, are different aspects of a unique Galois theory. The latter is based upon the construction and study of the tensor product of non commutative connections over a commutative base (semi-classical situation), without any curvature assumption. This theory provides for instance an algebraic frame for the study of the confluence arising when the increment of a difference equation tends to 0.

Computing Galois Groups of Completely Reducible Differential Equations

We give an algorithm to calculate a presentation of the Picard–Vessiot extension associated to a completely reducible linear differential equation (i.e. an equation whose Galois group is reductive). Using this, we show how to compute the Galois group of such an equation as well as properties of the Galois groups of general equations.

A new necessary condition for the integrability of Hamiltonian systems with a two-dimensional homogeneous potential

Recently, Morales-Ruiz and Ramis obtained a strong necessary condition for the integrability of Hamiltonian systems with a homogeneous potential based on their own theorem on the differential Galois theory (Picard–Vessiot theory) for Hamiltonian systems. The theorem claims that if the original Hamiltonian system is integrable, then the variational equation around a particular solution is solvable in the sense of the differential Galois theory, i.e., the solution is obtained only by a combination of quadratures, exponential of quadratures and algebraic functions. In this paper, a direct and independent proof of this statement is given for Hamiltonian systems with a two-dimensional homogeneous potential, which leads to the new necessary condition for integrability. This new necessary condition well justifies the so-called weak Painlevé conjecture of Ramani et al. for the first time.

Picard-Vessiot Theory and Ziglin′s Theorem

Given a Hamiltonian system, Ziglin′s theorem is one of the criteria used to study the integrability of the system. For this we make use of the variational equations along a known solution. In this paper we obtain some consequences of the application of Picard-Vessiot differential Galois theory to Ziglin′s theorem. First we present the general results for the case of a finite arbitrary number of degrees of freedom. Then we pass to the case of two degrees of freedom where the results can be given with more detail. Under the hypothesis of Ziglin′s theorem and some additional technical assumptions the main results (Theorems 1 and 2) relate the integrability of the Hamiltonian with the properties of the differential Galois group of the Picard Vessiot extension associated to the normal reduced variational equations. For two degrees of freedom it is possible to study also the resonant case. Theorems 4 and 5 give the full classification of the Galois groups and the related extensions in the integrable case. A couple of applications are made to recover Ito′s theorem and to study the Lamé equation.

Infeld–Hull factorization, Galois–Picard–Vessiot theory for differential operators

The Infeld–Hull theory of factorization of second order differential operators is related, on the one hand, to the classical Picard–Vessiot theory and, on the other hand, to work by the author, Gel’fand, and Kirillov on Lie algebras contained in the field of fractions or algebraic extensions of the enveloping associative algebra of a Lie algebra.