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Parallel telescoping is a natural generalization of differential creative-telescoping for single integrals to line integrals. It computes a linear ordinary differential operator $L$, called a parallel telescoper, for several multivariate…

We revisit the concept of special algebras, also known as \textit{purely inseparable ring extensions}. This concept extends the notion of purely inseparable field extensions to the more general context of extensions of commutative rings. We…

Over a non-closed field, it is a common strategy to use separable algebras as invariants to distinguish algebraic and geometric objects. The most famous example is the deep connection between Severi-Brauer varieties and central simple…

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…

A fundamental theorem of Katz \cite{Katz87} determines the differential Galois groups of rank $n$ connections on algebraic curves with slope $r/n$ at a singularity, where $\gcd(r,n)=1$. We extend this result to $G$-connections, where $G$ is…

Let $k$ be a number field. We provide an asymptotic formula for the number of Galois extensions of $k$ with absolute discriminant bounded by some $X \geq 1$, as $X\to\infty$. We also provide an asymptotic formula for the closely related…

The motivation for this paper is to extend the known model theoretic treatment of differential Galois theory to the case of linear difference equations (where the derivative is replaced by an automorphism.) The model theoretic difficulties…

The infinitesimal counterpart of a Lie groupoid is its Lie algebroid. As a vector bundle, it is given by the source vertical tangent bundle restricted to the identity bisection. Its sections can be identified with the invariant vector…

We study torsors for groups defined by algebraic difference equations. Our main result provides necessary and sufficient conditions on the base difference field for all such torsors to be trivial. We also present an application to the…

The problem of analyzing the number of number field extensions $L/K$ with bounded (relative) discriminant has been the subject of renewed interest in recent years, with significant advances made by Schmidt, Ellenberg-Venkatesh, Bhargava,…

The fundamental concepts in the Galois Theory are separable, normal and Galois field extensions. These concepts are central in proofs of the Galois Theory. In the paper, we introduce a new approach, a ring theoretic approach, to the Galois…

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq2$ with Galois automorphism $\sigma$, and let $R$ be an algebraically closed field of characteristic $\ell\notin\{0,p\}$. We…

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…

In this paper we compute the difference Galois groups of the Lie-irreducible regular singular generalized q-hypergeometric equations of order 3 with q-real parameters by using a density theorem due to Sauloy. In contrast with the…

Let $K$ be a local field of characteristic 0 with residue characteristic $p$. Let $G$ be an extraspecial $p$-group and let $L/K$ be a totally ramified $G$-extension. In this paper we find sufficient conditions for $L/K$ to admit a Galois…

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…

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…

We prove a Galois-type correspondence between compositions of purely inseparable field extensions (including infinite ones) and subalgebras of differential operators. This correspondence can be utilized to establish a connection between…

We extend Kovacic's algorithm to compute the differential Galois group of some second order parameterized linear differential equation. In the case where no Liouvillian solutions could be found, we give a necessary and sufficient condition…

We present a characterization of differential Picard-Vessiot extensions which is analogous to the field theoretic characterization of Galois extensions in classical Galois theory.