arXiv subjects
Publications and source records attributed to Lotfi Saidane.
We show that the determinant $\nabla(d,α),$ which seems to be not considered in the past, is not zero. As an application of this result we prove that the Setoyanagi operator $S_{p,q}=\partial^{2}-(ax^{p}+bx^{q}) $ is irreducible over $\mathbb{C}(x) [ \partial] $.
Let $(E)$ be a homogeneous linear differential equation Fuchsian of order $n$ over $\mathbb{P}^{1}(\mathbb{C}) $. The idea of Riemann (1857) was to obtain the properties of solutions of ($E$) by studying the local system. Thus, he obtained some properties of Gauss hypergeometric functions by studying the associated rank 2 local system over $\mathbb{P}^{1}(\mathbb{C}) \backslash\{3 points\} $. For example, he obtained the Kummer transformations of the hypergeometric functions without any calculation. The success of the Riemann's methods is due to the fact that the irreducible rank 2 local system over $\mathbb{P}^{1}(\mathbb{C}) \backslash\{3 points\} $ is linearly "rigid" in the sense of Katz \cite{Katz}. This result constitute one of the best studied example of linear rigid system, it was proved by the Levelt's theorem \cite{B} Theorem 1.2.3. In this work we propose a partial generalization of the Levelt's theorem.
Let $(E)$ a homogeneous linear differential equation of order $n$ Fuchsien over $\mathbb{P}^{1}(\mathbb{C}) $. The idea of Riemann (1857) was to obtain the properties of solutions of (E) by studying the local system. Thus, he obtained some properties of Gauss hypergeometric functions by studying the assocated rank 2 local system over $\mathbb{P}^{1}(\mathbb{C}) \backslash {3 points} $. For example, he obtained the Kummer transformations of the hypergeometric functions without any calculation. The success of the Riemann's methods is due to the fact that the irreducible rank 2 local system over $\mathbb{P}^{1}(\mathbb{C}) \backslash {3 points} $ are "rigid". Levelt theorem, see \cite{B} Theorem 1.2.3 proves this result. In this work, we propose a partial generalization of this theorem.
In this work we study a formal equivalent class of an Airy's D-module, which we compare with an analytic class of equivalence. We calculate the determining factors of an Airy operator of bidegree (n, m) and we clarify the coefficients of the operator which intervene in the determination of the determining factors. Finally, we induce, the canonical model (see [ BV1]), of an Airy operator of bidegre (n, m). ----- Dans ce travail on étudie une classe d'équivalence formelle d'un D-module d'Airy, qu'on compare avec une classe d'équivalence analytique. On calcule les facteurs déterminants d'un opérateur d'Airy de bidegré (n,m) et on précise les coefficients de l'opérateur qui interviennent dans la détermination des facteurs déterminants. On donne enfin, le modèle canonique, voir [BV1], d'un opérateur d'Airy de bidegré (n,m).