A family of holomorphic function spaces can be defined with reproducing kernels , obtained as real powers of the Cauchy-Szegö kernel. In this paper we study properties of the associated Poisson-like kernels: . In particular, we show boundedness of associated maximal operators, and obtain formulas for the limit of Poisson integrals in the topological boundary of the cone.
Using the notion of the maximal polar quotient we characterize the critical values at infinity of polynomials in two complex variables. As an application we give a necessary and sufficient condition for a family of affine plane curves to be equisingular at infinity.
Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in of codimension 1, . The lower bound is attained if and only if H is orthogonal to the versor of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify with ; by we denote the usual k-dimensional volume in . The result is a complex counterpart of Ball’s [B1] result for...
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.
Nous montrons comment calculer des équations fonctionnelles du type de Bernstein associées à une fonction et aux sections du module de cohomologie locale algébrique à support une intersection complète quasi-homogène à singularité isolée.
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 having singular points. As a function of , 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...