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By Mr (d: A and Pongérard Patrice


Sur l’existence d’une solution ramifiée pour des équations de Fuchs à caractéristique simple. (French. English summary) [Existence of a ramified solution for Fuchs equations with simple characteristic] Proc. Amer. Math. Soc. 137 (2009), no. 8, 2671–2683. Let (t, x) be a point of C × C n, and let Dt and D be differential operators with respect to t and x. The author considers the linear Fuchsian operator of order m ≥ 1 in a neighborhood of the origin O0 = D0 × Ω0 in C × C n of the form L = tA(t, x, Dt, D) + B(t, x, Dt, D). Here A and B are linear partial differential operators of order m and m − 1 and A has a simple characteristic hypersurface transverse to S: t = 0. Denote by T the hyperplane of S with the equation t = x1 = 0. It is possible to define a simple characteristic hypersurface K = k(t, x) in O0 going through T and transverse to S. For every δ> 0 denote by Dδ = {z ∈ C: |z | < δ} and by Rδ the universally ramified surface of a pointed circle. If v is a holomorphic function ramified around K, we can find such real δ> 0 and a connected neighborhood O ⊂ O0 of the origin in C × C n such that v(t, x) = v(z, t, x) | z=k(t,x). Under some conditions on the operator L the author gives a construction of a solution to the equation L(t, x, Dt, D)u(x, t) = v(z, t, x) | z=k(t,x). In addition, a natural condition is assumed for the characteristic polynomial connected to th

Topics: Reviewed by V. A. Golubeva
Year: 2013
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