Related papers: Regular singular differential equations and free p…
We determine the absolute differential Galois group of the field $\mathbb{C}(x)$ of rational functions: It is the free proalgebraic group on a set of cardinality $|\mathbb{C}|$. This solves a longstanding open problem posed by B.H. Matzat.…
We show that every linear algebraic group over an algebraically closed field of characteristic zero is the differential Galois group of a regular singular linear differential equation with rational function coefficients.
Let $C \langle t_1, \dots t_l\rangle$ be the differential field generated by $l$ differential indeterminates $\boldsymbol{t}=(t_1, \dots ,t_l)$ over an algebraically closed field $C$ of characteristic zero. We develop a lower bound…
We show that a closed finite index subgroup of a free proalgebraic group is itself a free proalgebraic group. Our main motivation for this result is an application in differential Galois theory: The absolute differential Galois group of a…
Let $G$ be a classical group of Lie rank $l$ and let $C$ be an algebraically closed field of characteristic zero. For $l$ differential indeterminates $\boldsymbol{v}=(v_1,\dots,v_l)$ over $C$ we constructed in a previous paper a general…
We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field $\mathbb{C}(x)$ with derivation $\frac{d}{dx}$ and endomorphism $f(x)\mapsto f(x+1)$. Our main result is…
We study families of linear differential equations parametrized by an algebraic variety $\mathcal{X}$ and show that the set of all points $x\in \mathcal{X}$, such that the differential Galois group at the generic fibre specializes to the…
We study the relation between the Galois group $G$ of a linear difference-differential system and two classes $\mathcal{C}_1$ and $\mathcal{C}_2$ of groups that are the Galois groups of the specializations of the linear difference equation…
Differential equations have arithmetic analogues in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations and the present paper is concerned with the "linear" ones. The equations…
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 develop algorithms to compute the differential Galois group corresponding to a one-parameter family of second order homogeneous ordinary linear differential equations with rational function coefficients. More precisely, we consider…
We give sufficient conditions for a linear differential equation to have a given semisimple group as its Galois group. For any linear algebraic group G given as a semidirect product of a finite subgroup and a normal subgroup that is a…
The aim of this paper is to give a new result of the differential Galois theory of linear ordinary differential equations. In particular, we compute differential Galois group for special type non-resonant Fuchsian system.
Differential Galois theory has played important roles in the theory of integrability of linear differential equation. In this paper we will extend the theory to nonlinear case and study the integrability of the first order nonlinear…
This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups.…
We study the interplay between the differential Galois group and the Lie algebra of infinitesimal symmetries of systems of linear differential equations. We show that some symmetries can be seen as solutions of a hierarchy of linear…
We study the structure of the absolute differential Galois group of a rational function field over an algebraically closed field of characteristic zero. In particular, we relate the behavior of differential embedding problems to the…
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…
Let $F$ be a characteristic zero differential field with algebraically closed constant field, and consider the compositum $F_u$ of all Picard--Vessiot extensions of $F$ with unipotent differential Galois group. We prove that the group of…
Let $C \langle \boldsymbol{t} \rangle$ be the differential field generated by $l$ differential indeterminates $\boldsymbol{t}=(t_1, \dots, t_l)$ over an algebraically closed field $C$ of characteristic zero. In this article we present an…