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On the Bernstein-Walsh-Siciak theorem

Rafał Pierzchała — 2012

Studia Mathematica

By the Oka-Weil theorem, each holomorphic function f in a neighbourhood of a compact polynomially convex set $K \subset ℂ^{N}$ can be approximated uniformly on K by complex polynomials. The famous Bernstein-Walsh-Siciak theorem specifies the Oka-Weil result: it states that the distance (in the supremum norm on K) of f to the space of complex polynomials of degree at most n tends to zero not slower than the sequence M(f)ρ(f)ⁿ for some M(f) > 0 and ρ(f) ∈ (0,1). The aim of this note is to deduce the uniform version,...

Invariance of domain in o-minimal structures

Rafał Pierzchała — 2001

Annales Polonici Mathematici

The aim of this paper is to prove the theorem on invariance of domain in an arbitrary o-minimal structure. We do not make use of the methods of algebraic topology and the proof is based merely on some basic facts about cells and cell decompositions.

On roots of polynomials with power series coefficients

Rafał Pierzchała — 2003

Annales Polonici Mathematici

We give a deepened version of a lemma of Gabrielov and then use it to prove the following fact: if h ∈ 𝕂[[X]] (𝕂 = ℝ or ℂ) is a root of a non-zero polynomial with convergent power series coefficients, then h is convergent.

On the Kuratowski convergence of analytic sets

Maciej P. Denkowski; Rafał Pierzchała — 2008

Annales Polonici Mathematici

We discuss some conditions which guarantee that the Kuratowski limit of a sequence of analytic sets is a Nash set.

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Currently displaying 1 – 4 of 4

On the Bernstein-Walsh-Siciak theorem

Invariance of domain in o-minimal structures

On roots of polynomials with power series coefficients

On the Kuratowski convergence of analytic sets