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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Sur l’existence d’une solution ramifiée pour des équations de Fuchs à caractéristique simple
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by Patrice Pongérard PDF
Proc. Amer. Math. Soc. 137 (2009), 2671-2683 Request permission


The aim of this paper is to construct a holomorphic solution, ramified around a simple characteristic hypersurface, for some linear Fuchsian equation of order $m\geq 1$. We consider an operator $L$, holomorphic in a neighborhood of the origin in ${\mathbb {C}}_t\times {\mathbb {C}}_x^n$, of the form $L=tA+B$ where $A$ and $B$ are linear partial differential operators of order $m$ and $m-1$, and where $A$ has a simple characteristic hypersurface transverse to $S:t=0$. Under an assumption linking the principal symbols of $A$ and $B$, the question is reduced to the study of an integro-differential Fuchsian equation with an additional variable $z$ that describes the universal covering of a pointed disk. It is an equation where terms like $t^lD_t^hD_x^\alpha (tD_t+1)^{-1}D_z^{-q}, l,h,q\in \mathbb {N}, \alpha \in \mathbb {N}^n$ with $l\leq 1$ and $h+|\alpha |\leq l+q$ appear. The problem is solved by the fixed-point theorem with appropriate estimations in a Banach space.
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Additional Information
  • Patrice Pongérard
  • Affiliation: Université de La Réunion, 23 allée des rubis, 97400 Saint-Denis, La Réunion, France
  • Email:
  • Received by editor(s): February 25, 2008
  • Received by editor(s) in revised form: October 16, 2008
  • Published electronically: February 3, 2009
  • Communicated by: Matthew J. Gursky
  • © Copyright 2009 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 2671-2683
  • MSC (2000): Primary 35A07; Secondary 35A20
  • DOI:
  • MathSciNet review: 2497480