相关论文: Hypertranscendance et Groupes de Galois aux differ…

Humans develop certain cognitive abilities to recognize objects and their transformations without explicit supervision, highlighting the importance of unsupervised representation learning. A fundamental challenge in unsupervised…

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,…

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 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…

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…

Generalizing Atiyah extensions, we introduce and study differential abelian tensor categories over differential rings. By a differential ring, we mean a commutative ring with an action of a Lie ring by derivations. In particular, these…

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 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…

In this paper, the changes of representations of a group are used in order to describe its action as algebraic Galois group of an univariate polynomial on the roots of factors of any Lagrange resolvent. By this way, the Galois group of…

We make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the…

After H\"older proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental i.e. they cannot be solution to an algebraic…

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 geometric setting for the differential Galois theory of $G$-invariant connections with parameters. As an application of some classical results on differential algebraic groups and Lie algebra bundles, we see that the Galois…

We develop a Galois theory for systems of linear difference equations with an action of an endomorphism {\sigma}. This provides a technique to test whether solutions of such systems satisfy {\sigma}-polynomial equations and, if yes, then…

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…

We give model theoretic accounts and proofs of the existence and uniqueness of differential Galois extensions with no new constants, for logarithmic differential equations over a differential field K, when the field C of constants of K is…

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…

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 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.…