Let D be a bounded domain in ℂn. A holomorphic function f: D → ℂ is called normal function if f satisfies a Lipschitz condition with respect to the Kobayashi metric on D and the spherical metric on the Riemann sphere ̅ℂ. We formulate and prove a few Lindelöf principles in the function theory of several complex variables.
This article is dedicated to the centenary of the local CR equivalence problem, formulated by Henri Poincaré in 1907. The first part gives an account of Poincaré’s heuristic counting arguments, suggesting existence of infinitely many local CR invariants. Then we sketch the beautiful completion of Poincaré’s approach to the problem in the work of Chern and Moser on Levi nondegenerate hypersurfaces. The last part is an overview of recent progress in solving the problem on Levi degenerate manifolds....
We prove that the study of the Łojasiewicz exponent at infinity of overdetermined polynomial mappings , m > n, can be reduced to the one when m = n.
The aim of this paper is to study the Łojasiewicz exponent of c-holomorphic mappings. After introducing an order of flatness for c-holomorphic mappings we give an estimate of the Łojasiewicz exponent in the case of isolated zero, which is a generalization of the one given by Płoski and earlier by Chądzyński for two variables.
We prove that the infimum of the regular separation exponents of two subanalytic sets at a point is a rational number, and it is also a regular separation exponent of these sets. Moreover, we consider the problem of attainment of this exponent on analytic curves.
Some estimates of the Łojasiewicz gradient exponent at infinity near any fibre of a polynomial in two variables are given. An important point in the proofs is a new Charzyński-Kozłowski-Smale estimate of critical values of a polynomial in one variable.
For every holomorphic function in two complex variables with an isolated critical point at the origin we consider the Łojasiewicz exponent ₀(f) defined to be the smallest θ > 0 such that near 0 ∈ ℂ² for some c > 0. We investigate the set of all numbers ₀(f) where f runs over all holomorphic functions with an isolated critical point at 0 ∈ ℂ².
We find a relation between the vanishing of a globally defined residue current on and solution of the membership problem with control of the polynomial degrees. Several classical results appear as special cases, such as Max Nöther’s theorem, for which we also obtain a generalization. Furthermore there are some connections to effective versions of the Nullstellensatz. We also provide explicit integral representations of the solutions.