Related papers: Descent for differential Galois theory of differen…
A Galois theory of differential fields with parameters is developed in a manner that generalizes Kolchin's theory. It is shown that all connected differential algebraic groups are Galois groups of some appropriate differential field…
We present a geometric setting for the differential Galois theory of $G$-invariant connections with parameters. As an application of some classical results on differential algebraic groups and Lie algebra bundles, we see that the Galois…
We establish some comparison results among the different parameterized Galois theories for $q$-difference equations, completing the work by CHatzidakis, Hardouin and Singer, that addresses the problem in the case without parameters. Our…
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…
In this paper, we study the algebraic relations satisfied by the solutions of $q$-difference equations and their transforms with respect to an auxiliary operator. Our main tool is the parametrized Galois theories developed in two papers.…
We develop a Galois theory for systems of linear difference equations with periodic parameters, for which we also introduce linear difference algebraic groups. We then apply this to constructively test if solutions of linear q-difference…
We propose in this paper a Galois theory of $q$-difference equations where q is a root of unity. This theory is the q difference analogue of the Galois theory of iterative differential equations, that is differential equations over fields…
We apply the differential Galois theory for difference equations developed by Hardouin and Singer to compute the differential Galois group for a second-order linear $q$-difference equation with rational function coefficients. This Galois…
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…
We study the differential Galois theory of difference equations under weaker hypothesis on the field of constants of the automorphism. This framework yields a new approach to results by C.Hardouin and M.Singer, which answers possitively a…
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,…
We develop a Galois theory for linear differential equations equipped with the action of an endomorphism. This theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation. The Galois…
We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups and we use structure theorems for these groups to…
Let C be an algebraically closed field and X a projective curve over C. Consider an ordinary linear differential equation, or a linear differ- ence equation, with coefficients in the field of rational functions of X, and assume that its…
This paper is on the inverse parameterized differential Galois problem. We show that surprisingly many groups do not occur as parameterized differential Galois groups over K(x) even when K is algebraically closed. We then combine the method…
This paper deals with criteria of algebraic independence for the derivatives of solutions of rank one difference equations. The key idea consists in deriving from the commutativity of the differentiation and difference operators a sequence…
If we consider a q-analogue of linear differential equation, Galoois group of the q-analogue difference equation is still a linear algebraic group. Namely, by a quantization of linear differential equation, Galois group is not quantized. We…
We show how the Galois-Picard_Vessiot theory of differential equations and difference equations, and the theory of holonomy groups in differential geometry, are different aspects of a unique Galois theory. The latter is based upon the…
Generalizing Atiyah extensions, we introduce and study differential abelian tensor categories over differential rings. By a differential ring, we mean a commutative ring with an action of a Lie ring by derivations. In particular, these…
This article is on the inverse Galois problem in Galois theory of linear iterative differential equations in positive characteristic. We show that it has an affirmative answer for reduced algebraic group schemes over any iterative…