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Galois theory of q-difference in the roots of unity

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Discrete Integrable Systems

For $q in mathbb{C}*$ non equal to $1$, we denote by $ igma_q$ the automorphism of $mathbb{C}(z)$ given by $ igma_q(f)(z)=f(qz)$. As q goes to $1$, a q-difference equation w.r.t. $ igma_q$ goes to a differential equation. The theory related to this fact is also called extit{confluence} and one part of its study is the behaviour of the related Galois groups during this process. Therefore it seems interresting to have a good Galois theory of q-difference equation for q equal to a root of unity. Because of the increasing size of the constant field at these points, such construction has been avoided for a long time. Recently P.Hendricks has proposed a solution to this problems but his Galois groups were defined over very transcendant fields. We propose here a new approach based, in a certain sense, on a q-deformation of the work of B.H. Matzat and Marius van der Put for Differential Galois theory in positive characteristic. We consider also a family of extit{iterative difference operator} instead of considering, just one difference operator, and by this way we stop the increasing of the constant field and succeed to set up a Picard-Vessiot Theory for q-difference equations where q is a root of unity and relate it to a Tannakian approach.

This talk is part of the Isaac Newton Institute Seminar Series series.

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