Related papers: A density theorem for the difference Galois groups…
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…
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 present a difference analogue of a result given by Hrushovski on differential Galois groups under specialization. Let $k$ be an algebraically closed field of characteristic zero and $\mathbb{X}$ an irreducible affine algebraic variety…
Galois theory is developed using elementary polynomial and group algebra. The method follows closely the original prescription of Galois, and has the benefit of making the theory accessible to a wide audience. The theory is illustrated by a…
The present paper essentially contains two results that generalize and improve some of the constructions of [arXiv:0801.1493]. First of all, in the case of one derivation, we prove that the parameterized Galois theory for difference…
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.…
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.…
We give a detailed proof of Kolchin's results on differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. We closely follow former works due to Pillay and…
We present a simple, but efficient, way to calculate connection matrices between sets of independent local solutions, defined at two neighboring singular points, of Fuchsian differential equations of quite large orders, such as those found…
This paper is devoted to the study of the analytic properties of Mahler systems at 0. We give an effective characterisation of Mahler systems that are regular singular at 0, that is, systems which are equivalent to constant ones. Similar…
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 present new computational results for symplectic monodromy groups of hypergeometric differential equations. In particular, we compute the arithmetic closure of each group, sometimes justifying arithmeticity. The results are obtained by…
An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on…
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…
The classical Galois theory deals with certain finite algebraic extensions and establishes a bijective order reversing correspondence between the intermediate fields and the subgroups of a group of permutations called the Galois group of…
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 develop the representation theory for reductive linear differential algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs, extending…
We describe a Picard-Vessiot theory for differential fields with non algebraically closed fields of constants. As a technique for constructing and classifying Picard-Vessiot extensions, we develop a Galois descent theory. We utilize this…
We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological…
We investigate for which linear-algebraic groups (over the complex numbers or any local field) there exists subgroups which are dense in the Zariski topology, but discrete in the Hausdorff topology. For instance, such subgroups exist for…