71 research outputs found
Differential Galois Theory of Linear Difference Equations
We present a Galois theory of difference equations designed to measure the
differential dependencies among solutions of linear difference equations. With
this we are able to reprove Hoelder's Theorem that the Gamma function satisfies
no polynomial differential equation and are able to give general results that
imply, for example, that no differential relationship holds among solutions of
certain classes of q-hypergeometric functions.Comment: 50 page
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
Iterative -Difference Galois Theory
Initially, the Galois theory of -difference equations was built for unequal to a root of unity. This choice was made in order to avoid the increase of the field of constants to a transcendental field. Inspired by the work of B.H. Matzat and M. van der Put, we consider in this paper a family of iterative difference operators instead of considering just one difference operator, and in this way we stop the increase of the constant field and succeed in setting up a Picard-Vessiot theory for -difference equations where is a root of unity that extend the Galois theory of difference equations of Singer and van der Put. The theory we obtain is quite the exact translation of the iterative differential Galois theory developed by B.H. Matzat and M. van der Put to the -difference world
On the Grothendieck conjecture on p-curvatures for q-difference equations
In the present paper, we give a q-analogue of the Grothendieck conjecture on
p-curvatures for q-difference equations defined over the field of rational
function K(x), where K is a finite extension of a field of rational functions
k(q), with k perfect. Then we consider the generic (also called intrinsic)
Galois group in the sense of N. Katz. The result in the first part of the paper
lead to a description of the generic Galois group through the properties of the
functional equations obtained specializing q on roots of unity. Although no
general Galois correspondence holds in this setting, in the case of positive
characteristic, where nonreduced groups appear, we can prove some devissage of
the generic Galois group.
In the last part of the paper, we give a complete answer to the analogue of
Grothendieck conjecture on -curvatures for q-difference equations defined
over the field of rational function K(x), where K is any finitely generated
extension of \mathbb Q and q\neq 0,1: we prove that the generic Galois group of
a q-difference module over K(x) always admits an adelic description in the
spirit of the Grothendieck-Katz conjecture. To this purpose, if q is an
algebraic number, we prove a generalization of the results by L. Di Vizio,
2002.Comment: The content of this paper was previously included in arXiv:1002.483
Calcul du groupe de Galois du produit de trois opérateurs différentiels complètement réductibles
Simplicity of the automorphism group of fields with operators
We adapt a proof of Lascar in order to show the simplicity of the group of
automorphisms fixing pointwise all non-generic elements for a class of
uncountable models of suitable theories of fields with operators
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