Méthodes symboliques pour les systèmes différentiels linéaires à singularité irrégulière

Abstract : This thesis is devoted to symbolic methods for local resolution of linear differential systems with coefficients in K = C((x)), the field of Laurent series, on an effective field C. More specifically, we are interested in effective algorithms for formal reduction. During the reduction, we are led to introduce algebraic extensions of the field of coefficients K (algebraic extensions of C, ramification of the variable x) in order to obtain a finer structure. From an algorithmic point of view, it is preferable to delay as much as possible the introduction of these extensions. To this end, we developed a new algorithm for formal reduction that uses the ring of endomorphisms of the system, called "eigenring". Using the formal classification given by Balser-Jurkat-Lutz, we deduce the structure of the eigenring of an indecomposable system. These theoretical results allow us to construct a decomposition on the base field K that separates the different exponential parts of the system and thus allows us to isolate, in indecomposable subsystems in K, the different algebraic extensions that can appear in order to treat them separately. In a second part, we are interested in Miyake’s algorithm for formal reduction. This algorithm is based on the computation of the Volevic weight and numbers of the valuation matrix of the system. We provide interpretations in graph theory and tropical algebra of the Volevic weight and numbers, and thus obtain practically efficient methods using linear programming. This completes a fundamental step in the Miyake reduction algorithm. These different algorithms are implemented as libraries for the computer algebra software Maple. Finally, we present a discussion on the performance of the reduction algorithm using the eigenring as well as a comparison in terms of timing between our implementation of Miyake’s reduction algorithm by linear programming and the algorithms of Barkatou and Pflügel.
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Joelle Saadé. Méthodes symboliques pour les systèmes différentiels linéaires à singularité irrégulière. Calcul formel [cs.SC]. Université de Limoges, 2019. Français. ⟨NNT : 2019LIMO0065⟩. ⟨tel-02444014⟩

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