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A KAM phenomenon for singular holomorphic vector fields

Laurent Stolovitch (2005)

Publications Mathématiques de l'IHÉS

Let X be a germ of holomorphic vector field at the origin of Cn and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection...

A variant of the complex Liouville-Green approximation theorem

Renato Spigler, Marco Vianello (2000)

Archivum Mathematicum

We propose a variant of the classical Liouville-Green approximation theorem for linear complex differential equations of the second order. We obtain rigorous error bounds for the asymptotics at infinity, in the spirit of F. W. J. Olver’s formulation, by using rather arbitrary ξ -progressive paths. This approach can provide higher flexibility in practical applications of the method.

Algebraic solutions of the Lamé equation

Frits Beukers (2002)

Banach Center Publications

In this paper we give a summary of joint work with Alexa van der Waall concerning Lamé equations having finite monodromy. This research is the subject of van der Waall's Ph. D. thesis [W].

Automorphic realization of residual Galois representations

Robert Guralnick, Michael Harris, Nicholas M. Katz (2010)

Journal of the European Mathematical Society

We show that it is possible in rather general situations to obtain a finite-dimensional modular representation ρ of the Galois group of a number field F as a constituent of one of the modular Galois representations attached to automorphic representations of a general linear group over F , provided one works “potentially.” The proof is based on a close study of the monodromy of the Dwork family of Calabi–Yau hypersurfaces; this in turn makes use of properties of rigid local systems and the classification...

Bernstein polynomials and spectral numbers for linear free divisors

Christian Sevenheck (2011)

Annales de l’institut Fourier

We discuss Bernstein polynomials of reductive linear free divisors. We define suitable Brieskorn lattices for these non-isolated singularities, and show the analogue of Malgrange’s result relating the roots of the Bernstein polynomial to the residue eigenvalues on the saturation of these Brieskorn lattices.

Classification rationnelle et confluence des systèmes aux différences singuliers réguliers

Julien Roques (2006)

Annales de l’institut Fourier

En choisissant des “caractères” et des “logarithmes”, méromorphes sur ℂ , construits à l’aide de la fonction Gamma d’Euler, et en utilisant des séries de factorielles convergentes, nous sommes en mesure, dans une première partie, de donner une “forme normale” pour les solutions d’un système aux différences singulier régulier. Nous pouvons alors définir une matrice de connexion d’un tel système. Nous étudions ensuite, suivant une idée de G.D. Birkhoff, le lien de celles-ci avec le problème de la classification...

Dynamique et formes normales d’équations différentielles implicites

Julien Aurouet (2014)

Annales de l’institut Fourier

Dans cet article on cherche à comprendre la dynamique locale d’équations différentielles implicites de la forme F ( x , y , d y ) = 0 , où F est un germe de fonction sur 𝕂 n × 𝕂 × 𝕂 n * (où 𝕂 = ℝ ou ℂ ), au voisinage d’un point singulier. Pour cela on utilise la relation intime entre les systèmes implicites et les champs liouvilliens. La classification par transformation de contact des équations implicites provient de la classification symplectique des champs liouvilliens. On utilise alors toute la théorie des formes normales pour les...

Euler's integral transformation for systems of linear differential equations with irregular singularities

Kouichi Takemura (2012)

Banach Center Publications

Dettweiler and Reiter formulated Euler's integral transformation for Fuchsian systems of differential equations and applied it to a definition of the middle convolution. In this paper, we formulate Euler's integral transformation for systems of linear differential equations with irregular singularities. We show by an example that the confluence of singularities is compatible with Euler's integral transformation.

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