Sur la compacité des ensembles de Julia des polynômes p-adiques

Let R be a rational function, of degree 2, with complex coefficients. Then the Julia set of R is a closed subset of ¹(), and therefore compact. If one replace by the field _p (completion of an algebraic closure of the field _p of the p-adic numbers), then one can define also a Julia set for a rational function with p-adic coefficients. But as _p is not locally compact, the Julia set may or may not be compact. In this paper, we study the compactness of the Julia set of p-adic polynomials. Mathematics Subject Classification (2000):11S99, 37B99.     

Une identité pour des polynômes d'Appell

In this paper, we establish an identity for some Appell polynomials generalizing explicit formulas for generalized Bernoulli numbers and polynomials.     

Construction de polynômes de Kazhdan-Lusztig arbitraires

To each polynomial P with integral non-negative coefficients and constant term equal to 1, of degree d, we associate a pair of elements (y,w) in the symmetric group S_n, where n = 1 + d + P(l), for which we prove that the Kazhdan-Lusztig polynomial P_y,w equals P. This pair satisfies ℓ(w) - ℓ(y) = 2d + P(1) - 1, where ℓ(w) denotes the number of inversions of w.     

Sur la densité d'état de l'opérateur de Schrödinger quasi-périodique unidimensionnel

We prove two results on the density of states of the discrete one dimensional quasi-periodic Schrödinger equation with an analytic potential and Diophantine frequencies in the perturbed regime. On the one hand, we prove that this function has the behavior of a Hölder-$\frac{1}{2}$ function. On the other, we show that the length of the gaps has a sub-exponential estimate which depends on its label given by the gap-labeling theorem. To cite this article: S. Hadj Amor, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Sur la fibre d'un polynôme à points critiques isolés

In this paper, we express the Euler characteristic of the fibers of a polynomial whose critical points are isolated in terms of signatures of appropriate bilinear symmetric forms. Received: 15 February 1999 / Revised version: 28 June 1999     

Configurations combinatoires des polynômes fibrés de degré 2

In this Note, we are interested in quadratic fibered polynomials which are skew products of the kind P_c : X × C → X × C, P_c(x, z) = (ƒ(x),z² + c(x)), where X is a compact set, ƒ is a continuous map from X into X, and c is a continuous complex valued function on X considered as a parameter. We construct a compact connected configuration space which gives a combinatorial model of a subset of the parameters space. Then, we explain in which way an abstract configuration can be realized by a quadratic fibered polynomial.     

Sur la détermination du périmètre de l'ovale à huit centres

This note aims to present the history of the main researches carried out to represent an ellipse by means of an eight-centered oval and to complete them by giving a geometrical demonstration of the formulas expressing the coordinates of the centers, the radii and the central angles of these arcs of circles making up this oval. These formulas will make it possible to determine its perimeter and thus to approximate the value of that of the ellipse. This result could be useful for archaeologists, who will test the hypothesis according to which some Roman amphitheatres, such as the Coliseum, were built from eight-centered ovals.     

Une propriété caractéristique des polynômes de Laguerre

Sans résumé     

Polynômes hyperboliques et combinaisons linéaires de certains polynômes hyperboliques

We give some results in the theory of hyperbolic polynomials and we study the hyperbolicity of some linear combinations of hyperbolic polynomials.Note présentée par Philippe G. Ciarlet.