Related papers: On the Galois group some Fuchsian systems
The difference Galois theory of Mahler equations is an active research area. The present paper aims at developing the analytic aspects of this theory. We first attach a pair of connection matrices to any regular singular Mahler equation. We…
The trajectory of energy is modeled by the solution of the Eikonal equation, which can be solved by solving a Hamiltonian system. This system is amenable of treatment from the point view of the theory of Differential Algebra. In particular,…
Computational Galois theory, in particular the problem of computing the Galois group of a given polynomial is a very old problem. Currently, the best algorithmic solution is Stauduhar's method. Computationally, one of the key challenges in…
We present an algorithm to determine the Galois group of an irreducible monic polynomial $f(x) \in \mathbb{Z}[x]$ of degree at most five. Following work of Conrad, Dummit, and Stauduhar this comes down to answering two questions: Is a given…
This note is a development of our two previous papers, arXiv:1212.3392v1 and 1306.3660v1. The fundamental question is whether there exists a Galois theory, in which the Galois group is a quantum group. For a linear equations with respect to…
For a differential operator $L$ of order $n$ over $C(z)$ with a finite (differential) Galois group $G\subset {\rm GL}(C^n)$, there is an algorithm, by M. van Hoeij and J.-A.~Weil, which computes the associated evaluation of the invariants…
In a previous work of the authors, a middle convolution operation on the category of Fuchsian differential systems was introduced. In this note we show that the middle convolution of Fuchsian systems preserves the property of global…
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,…
Lie group method provides an efficient tool to solve a differential equation. This paper suggests a fractional partner for fractional partial differential equations using a fractional characteristic method. A space-time fractional diffusion…
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…
For many finite groups, the Inverse Galois Problem can be approached through modular/automorphic Galois representations. This is a report explaining the basic strategy, ideas and methods behind some recent results. It focusses mostly on the…
The theme of this paper is to `solve' an absolutely irreducible differential module explicitly in terms of modules of lower dimension and finite extensions of the differential field $K$. Representations of semi-simple Lie algebras and…
We classify Galois objects for the dual of a group algebra of a finite group over an arbitrary field.
In this paper a Galoisian approach to build propagators through Riccati equations is presented. The main result corresponds to the relationship between the Galois integrability of the linear Schr\"odinger equation and the virtual…
This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generaliza- tion of linear systems, and the more general class of differential equations adimitting…
Group classification of systems of two coupled nonlinear reaction-diffusion equation with a diagonal diffusion matrix is carried out. Symmetries of diffusion systems with singular diffusion matrix and additional first order derivative terms…
We determine a considerable class of nonlinear partial differential equation systems which have global regular solutions. Uniqueness is not a direct general consequence of this method. The scheme can be applied to the incompressible Navier…
This note presents Galois theory for finite fields. It was written as a handout for the MAT401 course ``Polynomial equations and fields'' taught at the University of Toronto in Spring 2026. We use without proofs some basic properties of…
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…
We present a family of algorithms for computing the Galois group of a polynomial defined over a $p$-adic field. Apart from the "naive" algorithm, these are the first general algorithms for this task. As an application, we compute the Galois…