## 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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### References:

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