Generating operator of or Gaudin transfer matrices has quasi-exponential kernel.
Hyperbolic Cauchy problem and Leray's residue formula
We give an algebraic description of (wave) fronts that appear in strictly hyperbolic Cauchy problems. A concrete form of a defining function of the wave front issued from the initial algebraic variety is obtained with the aid of Gauss-Manin systems satisfied by Leray's residues.
Incompressibilité des feuilles de germes de feuilletages holomorphes singuliers
Nous considérons un germe de feuilletage holomorphe singulier non-dicritique défini sur une boule fermée , satisfaisant des hypothèses génériques, de courbe de séparatrice . Nous démontrons l’existence d’un voisinage ouvert de dans tel que, pour toute feuille de , l’inclusion naturelle induit un monomorphisme au niveau du groupe fondamental. Pour cela, nous introduisons la notion géométrique de « connexité feuilletée » avec laquelle nous réinterprétons la notion d’incompressibilité....
Intégrabilité et non-intégrabilité de systèmes hamiltoniens
Integrability of hamiltonian systems and differential Galois groups of higher variational equations
Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials
In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential , , of degree . The known results of Morales and Ramis give necessary conditions for the complete integrability of such systems. These conditions are expressed in terms of the eigenvalues of the Hessian matrix calculated at a non-zero point , such that . The main aim of this paper is to show that there are other obstructions for the integrability which appear if the matrix is not diagonalizable....
Kovalevska vs. Kovacic-two different notions of integrability and their connections
Ordinary differential equations all share the same common root-real physical problems. But, although the physical motivation remains the most important one, the way the subject develops does depend highly on the methods available. In the exposition I would like to show some connections between two methods of checking the ODE for integrability (whatever it should mean), with distant motivations and techniques. These are the so-called Painlevé tests and the methods originating in Ziglin's theory and...
Lamé operators with projective octahedral and icosahedral monodromies
Linearizability of a polynomial system
Matrices de Stokes-Ramis et constantes de connexion pour les systèmes différentiels linéaires de niveau unique
Etant donné un système différentiel linéaire de niveau unique quelconque, nous explicitons des formules donnant les multiplicateurs de Stokes en fonction de constantes de connexion dans le plan de Borel, généralisant ainsi les formules obtenues dans l’article Resurgence, Stokes phenomenon and alien derivatives for level-one linear differential systems (M. Loday-Richaud, P. Remy). Pour ce faire, nous nous ramenons à un système de niveaux par la méthode classique de réduction du rang ; puis, nous...
Middle convolution and Heun's equation.
Möbius transform for linear ordinary differential equations.
Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle
We consider germs of one-parameter generic families of resonant analytic diffeomorphims and we give a complete modulus of analytic classification by means of the unfolding of the Écalle modulus. We describe the parametric resurgence phenomenon. We apply this to give a complete modulus of orbital analytic classification for the unfolding of a generic resonant saddle of a 2-dimensional vector field by means of the unfolding of its holonomy map. Here again the modulus is an unfolding of the Martinet-Ramis...
Monodromic differential equations and Riemann theta functions.
Monodromie des systèmes différentiels linéaires à pôles simples sur la sphère de Riemann
New examples of holomorphic foliations without algebraic leaves
We present a series of polynomial planar vector fields without algebraic invariant curves in .
Normal forms for singularities of one dimensional holomorphic vector fields.
Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation
In this article, we study germs of holomorphic vector fields which are “higher order” perturbations of a quasihomogeneous vector field in a neighborhood of the origin of , fixed point of the vector fields. We define a “Diophantine condition” on the quasihomogeneous initial part which ensures that if such a perturbation of is formally conjugate to then it is also holomorphically conjugate to it. We study the normal form problem relatively to . We give a condition on that ensures that there...
Notes on algebraic functions.
On a “mysterious” case of a quadratic Hamiltonian.