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An invariant for difference field extensions

Zoé Chatzidakis, Ehud Hrushovski (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper we introduce a new invariant for extensions of difference fields, the distant degree, and discuss its properties.

Frobenius modules and Galois representations

B. Heinrich Matzat (2009)

Annales de l’institut Fourier

Frobenius modules are difference modules with respect to a Frobenius operator. Here we show that over non-archimedean complete differential fields Frobenius modules define differential modules with the same Picard-Vessiot ring and the same Galois group schemes up to extension by constants. Moreover, these Frobenius modules are classified by unramified Galois representations over the base field. This leads among others to the solution of the inverse differential Galois problem for p -adic differential...

Galois theory of q -difference equations

Marius van der Put, Marc Reversat (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

Choose q with 0 < | q | < 1 . The main theme of this paper is the study of linear q -difference equations over the field K of germs of meromorphic functions at 0 . A systematic treatment of classification and moduli is developed. It turns out that a difference module M over K induces in a functorial way a vector bundle v ( M ) on the Tate curve E q : = * / q that was known for modules with ”integer slopes“, [Saul, 2]). As a corollary one rediscovers Atiyah’s classification ( [ A t ] ) of the indecomposable vector bundles on the complex Tate...

On a general difference Galois theory I

Shuji Morikawa (2009)

Annales de l’institut Fourier

We know well difference Picard-Vessiot theory, Galois theory of linear difference equations. We propose a general Galois theory of difference equations that generalizes Picard-Vessiot theory. For every difference field extension of characteristic 0 , we attach its Galois group, which is a group of coordinate transformation.

On a general difference Galois theory II

Shuji Morikawa, Hiroshi Umemura (2009)

Annales de l’institut Fourier

We apply the General Galois Theory of difference equations introduced in the first part to concrete examples. The General Galois Theory allows us to define a discrete dynamical system being infinitesimally solvable, which is a finer notion than being integrable. We determine all the infinitesimally solvable discrete dynamical systems on the compact Riemann surfaces.

Picard-Vessiot theory in general Galois theory

Hiroshi Umemura (2011)

Banach Center Publications

We give a transparent proof that difference Picard-Vessiot theory is a part of the general difference Galois theory. We apply the proof to iterative q-difference Picard-Vessiot theory to show that Picard-Vessiot theory for iterative q-difference field extensions is in the scope of the general Galois theory of Heiderich. We also show that Picard-Vessiot theory is commutative in the sense that studying linear difference-differential equations, no matter how twisted the operators are, we cannot encounter...

Polynômes de Barsky

Youssef Haouat, Fulvio Grazzini (1979)

Annales scientifiques de l'Université de Clermont. Mathématiques

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