Related papers: Large Fields in Differential Galois Theory

We study integral points on varieties with infinite \'etale fundamental groups. More precisely, for a number field $F$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $\varphi\colon Y \to X$ of degree…

We point out the relevance of the Differential Galois Theory of linear differential equations for the exact semiclassical computations in path integrals in quantum mechanics. The main tool will be a necessary condition for complete…

A main problem in Galois theory is to characterize the fields with a given absolute Galois group. We apply a K-theoretic method for constructing valuations to study this problem in various situations. As a first application we obtain an…

We construct a Galois correspondence for finite purely inseparable field extensions $F/K$, generalising a classical result of Jacobson for extensions of exponent one (where $x^p \in K$ for all $x\in F$).

We show that each connected group scheme of finite type over an arbitrary ground field is isomorphic to the component of the identity inside the automorphism group scheme of some projective, geometrically integral scheme. The main…

We introduce the notion of differential largeness for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of "tame" differential fields. We state several…

We carry on the approach used in [Sch] to provide a solution for the inverse problem of the calculus of variations for Maxwell equations in vacuum and we provide an abstract theory including all implicit differential equations that can be…

We study a necessary condition for the integrability of the polynomials fields in the plane by means of the differential Galois theory. More concretely, by means of the variational equations around a particular solution it is obtained a…

Let $G$ be one of the classical groups of Lie rank $l$. We make a similar construction of a general extension field in differential Galois theory for $G$ as E. Noether did in classical Galois theory for finite groups. More precisely, we…

This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the…

We introduce a notion of inertial equivalence for integral $\ell$-adic representation of the Galois group of a global field. We show that the collection of continuous, semisimple, pure $\ell$-adic representations of the absolute Galois…

This article is the second part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part is concerned with symplectic Galois representations having…

I begin from a particular field of generalised Puiseux series and investigate a class of nonlinear differential equations in the field. It is appeared that the main part of differential equation determines solvability and positions of…

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…

Let $G$ be a finite classical group of Lie type of rank $\ell$, defined over a field of characteristic $p>2$. In this work, we classify the irreducible representations of $G$ whose dimensions are bounded by a constant proportional to…

As a simple corollary of a highly general framework for differential and difference Galois theory introduced by Y. Andre, we formulate a version of the Galois correspondence that applies over a difference field with arbitrary field of…

We prove that the main examples in the theory of algebraic differential equations possess a remarkable total differential overconvergence property. This allows one to consider solutions to these equations with coordinates in algebraically…

We consider a quantum theory based on a Galois field. In this approach infinities cannot exist, the cosmological constant problem does not arise, and one irreducible representation (IR) of the symmetry algebra splits into independent IRs…

Full set of autonomous completely solvable differential systems of equations in total differentials is built by basis of infinitesimal operators, universal invariant, and structure constants of admited multiparametric Lie group (abelian and…

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…