3,749 research outputs found
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
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
Darboux evaluations of algebraic Gauss hypergeometric functions
This paper presents explicit expressions for algebraic Gauss hypergeometric
functions. We consider solutions of hypergeometric equations with the
tetrahedral, octahedral and icosahedral monodromy groups. Conceptually, we
pull-back such a hypergeometric equation onto its Darboux curve so that the
pull-backed equation has a cyclic monodromy group. Minimal degree of the
pull-back coverings is 4, 6 or 12 (for the three monodromy groups,
respectively). In explicit terms, we replace the independent variable by a
rational function of degree 4, 6 or 12, and transform hypergeometric functions
to radical functions.Comment: The list of seed hypergeometric evaluations (in Section 2) reduced by
half; uniqueness claims explained; 34 pages; Kyushu Journal of Mathematics,
201
Zariski Closures of Reductive Linear Differential Algebraic Groups
Linear differential algebraic groups (LDAGs) appear as Galois groups of
systems of linear differential and difference equations with parameters. These
groups measure differential-algebraic dependencies among solutions of the
equations. LDAGs are now also used in factoring partial differential operators.
In this paper, we study Zariski closures of LDAGs. In particular, we give a
Tannakian characterization of algebraic groups that are Zariski closures of a
given LDAG. Moreover, we show that the Zariski closures that correspond to
representations of minimal dimension of a reductive LDAG are all isomorphic. In
addition, we give a Tannakian description of simple LDAGs. This substantially
extends the classical results of P. Cassidy and, we hope, will have an impact
on developing algorithms that compute differential Galois groups of the above
equations and factoring partial differential operators.Comment: 26 pages, more detailed proof of Proposition 4.
The q-analogue of the wild fundamental group and the inverse problem of the Galois theory of q-difference equations
In previous papers, we defined -analogues of alien derivations for linear
analytic -difference equations with integral slopes and proved a density
theorem (in the Galois group) and a freeness theorem. In this paper, we
completely describe the wild fundamental group and apply this result to the
inverse problem in -difference Galois theory. The new version contains an
appendix on pronilpotent completion and the main result on the direct problem
is made more precise. (Submitted for publication
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
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