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The canonical filtration by the slopes of a \(q\)-difference module and the associated graded module. (La filtration canonique par les pentes d’un module aux \(q\)-différences et le gradué associé.) (French) Zbl 1061.39013
The author presents the Newton polygon for a \(q\)-difference operator and for a \(q\)-difference module. The Newton polygon is constructed for a \(q\)-difference module and its intrinsic character is proved. The main ingredient is the Jordan-Hölder Theorem. The asymptotic behavior of this polygon with respect to the linear operations is described.
Next, the author studies the sub-module of the maximum range of the given slope and shows the existence of the canonical filtration. This is a translation of the Birkhoff-Guenter factorization Theorem. The filtration and the associated graded module have some important properties regarding the linear operations.

MSC:
39A13 Difference equations, scaling (\(q\)-differences)
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