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SUMMARY:Galois theory of q-difference in the roots of unity - Hardouin\, C
  (Heidelberg)
DTSTART:20090515T083000Z
DTEND:20090515T093000Z
UID:TALK18399@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION: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 different
 ial equation. The theory related to this fact is also called 	extit{conflu
 ence} and one part of its study is the behaviour of the related Galois gro
 ups during this process. Therefore it seems interresting to have a good Ga
 lois theory of q-difference equation for q equal to a root of unity. Becau
 se of the increasing size of the constant field at these points\, such con
 struction has been avoided for a long time. Recently P.Hendricks has propo
 sed a solution to this problems but his Galois groups were defined over ve
 ry transcendant fields. We propose here a new approach based\, in a certai
 n 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 conside
 r also a family of 	extit{iterative difference operator} instead of consid
 ering\, just one difference operator\, and by this way we stop the increas
 ing of the constant field and succeed to set up a Picard-Vessiot Theory fo
 r q-difference equations where q is a root of unity and relate it to a Tan
 nakian approach.
LOCATION:Satellite
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