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Index of a first order \(p\)-adic differential operator. IV: The case of systems. Measures of irregularity in a ball. (Indice d’un opérateur différentiel \(p\)-adique. IV: Cas des systèmes. Mesure de l’irrégularité dans un disque.) (French) Zbl 0548.12016

We are interested in showing that the differential operator of order \(1\), \({d\over dx}+G\), where \(G\) is a \(k\times k\) matrix with rational coefficients, has an index in the space of functions analytic in a ball; we then wish to compute this index. In the case \(k=1\), we show that this index exists (provided that the exponent of the differential operator in the ball is not a \(p\)-adic Liouville number) and we indicate how to compute this index. We can also show existence of index and compute this index when the system is equivalent to a triangular system. We give an interpretation of that index in term of the global irregularity of the differential operator in the ball.
For Part III, see the preceding review Zbl 0548.12015.
Reviewer: Philippe Robba

MSC:

12H25 \(p\)-adic differential equations
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
47E05 General theory of ordinary differential operators
14F30 \(p\)-adic cohomology, crystalline cohomology

Citations:

Zbl 0548.12015
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References:

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