614 research outputs found
Computing the differential Galois group of a parameterized second-order linear differential equation
We develop algorithms to compute the differential Galois group associated
to a parameterized second-order homogeneous linear differential equation of the
form where the coefficients are rational
functions in with coefficients in a partial differential field of
characteristic zero. Our work relies on the procedure developed by Dreyfus to
compute under the assumption that . We show how to complete this
procedure to cover the cases where , by reinterpreting a classical
change of variables procedure in Galois-theoretic terms.Comment: 14 page
Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation
We propose a new method to compute the unipotent radical of the
differential Galois group associated to a parameterized second-order
homogeneous linear differential equation of the form
where is a rational
function in with coefficients in a -field of characteristic zero,
and is a commuting set of parametric derivations. The procedure developed
by Dreyfus reduces the computation of to solving a creative
telescoping problem, whose effective solution requires the assumption that the
maximal reductive quotient is a -constant linear differential
algebraic group. When this condition is not satisfied, we compute a new set of
parametric derivations such that the associated differential Galois
group has the property that is -constant, and such
that is defined by the same differential equations as . Thus
the computation of is reduced to the effective computation of
. We expect that an elaboration of this method will be successful in
extending the applicability of some recent algorithms developed by Minchenko,
Ovchinnikov, and Singer to compute unipotent radicals for higher order
equations.Comment: 12 page
A density theorem for parameterized differential Galois theory
We study parameterized linear differential equations with coefficients
depending meromorphically upon the parameters. As a main result, analogously to
the unparameterized density theorem of Ramis, we show that the parameterized
monodromy, the parameterized exponential torus and the parameterized Stokes
operators are topological generators in Kolchin topology, for the parameterized
differential Galois group introduced by Cassidy and Singer. We prove an
analogous result for the global parameterized differential Galois group, which
generalizes a result by Mitschi and Singer. These authors give also a necessary
condition on a group for being a global parameterized differential Galois
group; as a corollary of the density theorem, we prove that their condition is
also sufficient. As an application, we give a characterization of completely
integrable equations, and we give a partial answer to a question of Sibuya
about the transcendence properties of a given Stokes matrix. Moreover, using a
parameterized Hukuhara-Turrittin theorem, we show that the Galois group
descends to a smaller field, whose field of constants is not differentially
closed.Comment: To appear in Pacific Journal of Mathematic
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
Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups
We present a Galois theory of parameterized linear differential equations
where the Galois groups are linear differential algebraic groups, that is,
groups of matrices whose entries are functions of the parameters and satisfy a
set of differential equations with respect to these parameters. We present the
basic constructions and results, give examples, discuss how isomonodromic
families fit into this theory and show how results from the theory of linear
differential algebraic groups may be used to classify systems of second order
linear differential equations
Tannakian categories, linear differential algebraic groups, and parameterized linear differential equations
We provide conditions for a category with a fiber functor to be equivalent to
the category of representations of a linear differential algebraic group. This
generalizes the notion of a neutral Tannakian category used to characterize the
category of representations of a linear algebraic group.Comment: 26 pages; corrected misprints; simplified Definition 2; more
references adde
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