相关论文: Th\'eorie analytique locale des \'equations aux $q…

This paper describes the classification of analytic $q$-difference equations. The difference Galois groups are computed. A tentative description of the universal difference Galois group is given.

We study the relationship between the local and global Galois theory of function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and…

In Proposition I of "Memoire sur les conditions de resolubilite des equations par radicaux", Galois established that any intermediate extension of the splitting field of a polynomial with rational coefficients is the fixed field of its…

We establish some comparison results among the different parameterized Galois theories for $q$-difference equations, completing the work by CHatzidakis, Hardouin and Singer, that addresses the problem in the case without parameters. Our…

If we consider a q-analogue of linear differential equation, Galoois group of the q-analogue difference equation is still a linear algebraic group. Namely, by a quantization of linear differential equation, Galois group is not quantized. We…

We develop a Galois theory for systems of linear difference equations with periodic parameters, for which we also introduce linear difference algebraic groups. We then apply this to constructively test if solutions of linear q-difference…

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 local analytic classification of irregular linear q-difference equations (Ramis-Sauloy-Zhang) involves the classfication of filtered q-difference modules with a prescribed associated graded module. We prove in a more general setting the…

The local analytic classification of irregular linear q-difference equations has recently been obtained by J.-P. Ramis, J. Sauloy and C. Zhang. Their description involves a q-analog of the Stokes sheaf and theorems of Malgrange-Sibuya type…

We compare several definitions of the Galois group of a linear difference equation that have arisen in algebra, analysis and model theory and show, that these groups are isomorphic over suitable fields. In addition, we study properties of…

Grothendieck's conjecture on p-curvatures predicts that an arithmetic differential equation has a full set of algebraic solutions if and only if its reduction in positive characteristic has a full set of rational solutions for almost all…

We study the distribution of the Galois group of a random $q$-additive polynomial over a rational function field: For $q$ a power of a prime $p$, let $f=X^{q^n}+a_{n-1}X^{q^{n-1}}+\ldots+a_1X^q+a_0X$ be a random polynomial chosen uniformly…

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…

We present an algorithm that determines the Galois group of linear difference equations with rational function coefficients.

Let $K$ be the function field of a smooth projective geometrically integral curve over a finite extension of $\mathbb{Q}_p$. Following the works of Harari, Scheiderer, Szamuely, Izquierdo, and Tian, we study the local-global and weak…

In this paper, we prove the "local epsilon-isomorphism conjecture" of Fukaya and Kato for a particular class of Galois modules obtained by tensoring a Zp-lattice in a crystalline representation of the Galois group of Qp with a…

We prove local-global compatibility results at $\ell=p$ for the torsion automorphic Galois representations constructed by Scholze, generalising the work of Caraiani--Newton. In particular, we verify, up to a nilpotent ideal, 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 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…

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…