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Polynomial bounds for the oscillation of solutions of Fuchsian systems

Gal Binyamini, Sergei Yakovenko (2009)

Annales de l’institut Fourier

We study the problem of placing effective upper bounds for the number of zeroes of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of a given dimension n having m singular points. As a function of n , m , this bound turns out to be double exponential in the precise sense explained in the paper.As a corollary, we obtain a solution of the so-called restricted infinitesimal...

Positivity, vanishing theorems and rigidity of Codimension one Holomorphic Foliations

O. Calvo-Andrade (2009)

Annales de la faculté des sciences de Toulouse Mathématiques

It is a known fact that the space of codimension one holomorphic foliations with singularities with a given ‘normal bundle’ has a natural structure of an algebraic variety. The aim of this paper is to consider the problem of the description of its irreducible components. To do this, we are interested in the problem of the existence of an integral factor of a twisted integrable differential 1–form defined on a projective manifold. We are going to do a geometrical analysis of the codimension one foliation...

Premiers pas en calcul étranger

B. Candelpergher, Jean-Claude Nosmas, Frédéric Pham (1993)

Annales de l'institut Fourier

Cet exposé est une introduction au calcul étranger d’Écalle, c’est-à-dire au calcul des obstructions à la sommabilité de Borel d’une grande classe de séries formelles, les fonctions résurgentes d’Écalle. La théorie d’Écalle éclaire d’un jour neuf le célèbre phénomène de Stokes qui est illustré ici dans le contexte de la méthode du col.

Puiseux Expansion of a Cuspidal Singularity

Maciej Borodzik (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

We present an effective and elementary method of determining the topological type of a cuspidal plane curve singularity with given local parametrization.

Puiseux series polynomial dynamics and iteration of complex cubic polynomials

Jan Kiwi (2006)

Annales de l’institut Fourier

We let 𝕃 be the completion of the field of formal Puiseux series and study polynomials with coefficients in 𝕃 as dynamical systems. We give a complete description of the dynamical and parameter space of cubic polynomials in 𝕃 [ ζ ] . We show that cubic polynomial dynamics over 𝕃 and ℂ are intimately related. More precisely, we establish that some elements of 𝕃 naturally correspond to the Fourier series of analytic almost periodic functions (in the sense of Bohr) which parametrize (near infinity) the quasiconformal...

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