5,189 research outputs found
Parameterized generic Galois groups for q-difference equations, followed by the appendix "The Galois D-groupoid of a q-difference system" by Anne Granier
We introduce the parameterized generic Galois group of a q-difference module,
that is a differential group in the sense of Kolchin. It is associated to the
smallest differential tannakian category generated by the q-difference module,
equipped with the forgetful functor. Our previous results on the Grothendieck
conjecture for q-difference equations lead to an adelic description of the
parameterized generic Galois group, in the spirit of the Grothendieck-Katz's
conjecture on p-curvatures. Using this description, we show that the
Malgrange-Granier D-groupoid of a nonlinear q-difference system coincides, in
the linear case, with the parameterized generic Galois group introduced here.
The paper is followed by an appendix by A. Granier, that provides a quick
introduction to the D-groupoid of a non-linear q-difference equation.Comment: The content of this paper was previously included in arXiv:1002.483
Variations for Some Painlev\'e Equations
This paper first discusses irreducibility of a Painlev\'e equation . We
explain how the Painlev\'e property is helpful for the computation of special
classical and algebraic solutions. As in a paper of Morales-Ruiz we associate
an autonomous Hamiltonian to a Painlev\'e equation . Complete
integrability of is shown to imply that all solutions to are
classical (which includes algebraic), so in particular is solvable by
''quadratures''. Next, we show that the variational equation of at a given
algebraic solution coincides with the normal variational equation of
at the corresponding solution. Finally, we test the Morales-Ramis
theorem in all cases to where algebraic solutions are present,
by showing how our results lead to a quick computation of the component of the
identity of the differential Galois group for the first two variational
equations. As expected there are no cases where this group is commutative
Galois differential algebras and categorical discretization of dynamical systems
A categorical theory for the discretization of a large class of dynamical
systems with variable coefficients is proposed. It is based on the existence of
covariant functors between the Rota category of Galois differential algebras
and suitable categories of abstract dynamical systems. The integrable maps
obtained share with their continuous counterparts a large class of solutions
and, in the linear case, the Picard-Vessiot group.Comment: 19 pages (examples added
A Reduced Form for Linear Differential Systems and its Application to Integrability of Hamiltonian Systems
Let with be a differential linear
system. We say that a matrix is a {\em reduced
form} of if and there exists such that . Such a form is
often the sparsest possible attainable through gauge transformations without
introducing new transcendants. In this article, we discuss how to compute
reduced forms of some symplectic differential systems, arising as variational
equations of hamiltonian systems. We use this to give an effective form of the
Morales-Ramis theorem on (non)-integrability of Hamiltonian systems.Comment: 28 page
Galois theory of fuchsian q-difference equations
We propose an analytical approach to the Galois theory of singular regular
linear q-difference systems. We use Tannaka duality along with Birkhoff's
classification scheme with the connection matrix to define and describe their
Galois groups. Then we describe \emph{fundamental subgroups} that give rise to
a Riemann-Hilbert correspondence and to a density theorem of Schlesinger's
type.Comment: Prepublication du Laboratoire Emile Picard n.246. See also
http://picard.ups-tlse.f
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