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Dimension de l’espace des solutions de l’équation \(y'=fy\) sur un infraconnexe et prolongement à travers un T-filtre. (Dimension of the solution space of the equation \(y'=fy\) on an infra-connection and continuation across a T-filter). (French) Zbl 0709.12005

Sémin. Anal., Univ. Blaise Pascal 1987-1988, No. 1 & 10, 15 p. (1988).
Let K be an algebraically closed field of characteristic zero which is complete under a non-archimedean valuation. Let D be a bounded set in K and let H(D) denote the set of Krasner’s analytic elements on D (i.e. the closure for uniform convergence of the set of rational functions without poles in D). Let f be any function in H(D). The aim of this paper is to study the space of solutions of the differential equation \(y'=fy\) in H(D). The most surprising result is that this space can be of any finite dimension for some “not to much irregular” (namely quasi-invertible) f. The result that should be the most useful is a necessary and sufficient condition for the existence of non-trivial solutions in the case where D is a closed disk. Proofs are not given but there is a complete bibliography.
Reviewer: G.Christol

MSC:

12H25 \(p\)-adic differential equations
12J27 Krasner-Tate algebras
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)