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Pinceaux de courbes planes et invariants polaires

Evelia R. García Barroso, Arkadiusz Płoski (2004)

Annales Polonici Mathematici

We study pencils of plane curves f t = f - t l N , t ∈ ℂ, using the notion of polar invariant of the plane curve f = 0 with respect to a smooth curve l = 0. More precisely we compute the jacobian Newton polygon of the generic fiber f t , t ∈ ℂ. The main result gives the description of pencils which have an irreducible fiber. Furthermore we prove some applications of the local properties of pencils to singularities at infinity of polynomials in two complex variables.

Plurisubharmonic functions with logarithmic singularities

E. Bedford, B. A. Taylor (1988)

Annales de l'institut Fourier

To a plurisubharmonic function u on C n with logarithmic growth at infinity, we may associate the Robin function ρ u ( z ) = lim sup λ → ∞ u ( λ z ) - log ( λ z ) defined on P n - 1 , the hyperplane at infinity. We study the classes L + , and (respectively) L p of plurisubharmonic functions which have the form u = log ( 1 + | z | ) + O ( 1 ) and (respectively) for which the function ρ u is not identically - ∞ . We obtain an integral formula which connects the Monge-Ampère measure on the space C n with the Robin function on P n - 1 . As an application we obtain a criterion on the convergence of the Monge-Ampère...

Polynôme d'Alexander à l'infini d'un polynôme à deux variables.

Enrique Artal Bartolo, Pierrette Cassou-Noguès (2000)

Revista Matemática Complutense

In this work, we compute the Alexander invariants at infinity of a complex polynomial in two variables by means of its resolution and also by means of the Eisenbud-Neumann diagram of the generic link at infinity of the polynomial.

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.

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