631 research outputs found
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
Real difference Galois theory
In this paper, we develop a difference Galois theory in the setting of real
fields. After proving the existence and uniqueness of the real Picard-Vessiot
extension, we define the real difference Galois group and prove a Galois
correspondence.Comment: Final versio
Confluence of meromorphic solutions of q-difference equations
In this paper, we consider a q-analogue of the Borel-Laplace summation where
q>1 is a real parameter. In particular, we show that the Borel-Laplace
summation of a divergent power series solution of a linear differential
equation can be uniformly approximated on a convenient sector, by a meromorphic
solution of a corresponding family of linear q-difference equations. We perform
the computations for the basic hypergeometric series. Following J. Sauloy, we
prove how a fundamental set of solutions of a linear differential equation can
be uniformly approximated on a convenient domain by a fundamental set of
solutions of a corresponding family of linear q-difference equations. This
leads us to the approximations of Stokes matrices and monodromy matrices of the
linear differential equation by matrices with entries that are invariants by
the multiplication by q
Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions
We develop general criteria that ensure that any non-zero solution of a given
second-order difference equation is differentially transcendental, which apply
uniformly in particular cases of interest, such as shift difference equations,
q-dilation difference equations, Mahler difference equations, and elliptic
difference equations. These criteria are obtained as an application of
differential Galois theory for difference equations. We apply our criteria to
prove a new result to the effect that most elliptic hypergeometric functions
are differentially transcendental
Differential algebraic generating series of weighted walks in the quarter plane
In the present paper we study the nature of the trivariate generating series
of weighted walks in the quarter plane. Combining the results of this paper to
previous ones, we complete the proof of the following theorem. The series
satisfies a nontrivial algebraic differential equation in one of its variable,
if and only if it satisfies a nontrivial algebraic differential equation in
each of its variables
Computing the Galois group of some parameterized linear differential equation of order two
We extend Kovacic's algorithm to compute the differential Galois group of
some second order parameterized linear differential equation. In the case where
no Liouvillian solutions could be found, we give a necessary and sufficient
condition for the integrability of the system. We give various examples of
computation.Comment: 14 pages, final version. To appear in Proceedings of the American
Mathematical Societ
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