Vorschau

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Abstract

Initially, the Galois theory of $q$-difference equations was built for $q$ unequal to a root of unity. This choice was made in order to avoid the increase of the field of constants to a transcendental field. Inspired by the work of B.H. Matzat and M. van der Put, we consider in this paper a family of iterative difference operators instead of considering just one difference operator, and in this way we stop the increase of the constant field and succeed in setting up a Picard-Vessiot theory for $q$-difference equations where $q$ is a root of unity that extend the Galois theory of difference equations of Singer and van der Put. The theory we obtain is quite the exact translation of the iterative differential Galois theory developed by B.H. Matzat and M. van der Put to the $q$-difference world.