Etude algébrique et algorithmique des singularités des équations différentielles implicites

Abstract : The set of solutions of an algebraic differential equation splits into the general solution and the singular solutions. These two concepts are given a precise meaning in the setting of differential algebra, a theory initiated by J.F.Ritt. Recent progresses in that field have brought effective algorithms that decide of the triviality of a differential system and provide a first decomposition of the set of solutions. We can thus determine if a differential equation admits any singular solutions and describe those. The decompositions there obtained are nonetheless not minimal. In this memoir, we propose a factorization free algorithm that eliminates the redundant components. The analytic interpretation is that we split the set of singular solutions into the singular solutions that are envelopes of the general solution and the singular solutions which are the limits of some other solutions. are envelopes of the general solution and the singular solutions which are the limits of some other solutions. The crux of the algorithm lies in the Low power theorem, while the effective implementation is based on the Rosenfeld-Gröbner algorithm. We furthermore present an algorithm and some criteria to compute the differential bases of the essential components of a differential equation. Such bases allow an analysis of singularities and integration heuristics.
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Evelyne Hubert. Etude algébrique et algorithmique des singularités des équations différentielles implicites. Modélisation et simulation. Institut National Polytechnique de Grenoble - INPG, 1997. Français. ⟨tel-00004947⟩

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