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On roots of polynomials with power series coefficients

100%
Pierzchała R.
Annales Polonici Mathematici
|
2003
|
tom 80
|
nr 1
211-217
EN
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.
2
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Invariance of domain in o-minimal structures

100%
Pierzchała R.
Annales Polonici Mathematici
|
2001
|
tom 77
|
nr 3
289-294
EN
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.
3
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On the Bernstein-Walsh-Siciak theorem

100%
Pierzchała R.
Studia Mathematica
|
2012
|
tom 212
|
nr 1
55-63
EN
By the Oka-Weil theorem, each holomorphic function f in a neighbourhood of a compact polynomially convex set $K ⊂ ℂ^{N}$ 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, sometimes called family version, of the Bernstein-Walsh-Siciak theorem, which is due to Pleśniak, directly from its classical (weak) form. Our method, involving the Baire category theorem in Banach spaces, appears to be useful also in a completely different context, concerning Łojasiewicz's inequality.
4
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On the Kuratowski convergence of analytic sets

63%
Denkowski M., Pierzchała R.
Annales Polonici Mathematici
|
2008
|
tom 93
|
nr 2
101-112
EN
We discuss some conditions which guarantee that the Kuratowski limit of a sequence of analytic sets is a Nash set.
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