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1
Content available remote

A counterexample to subanalyticity of an arc-analytic function

100%
Kurdyka K.
Annales Polonici Mathematici
|
1991
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tom 55
|
nr 1
241-243
EN
We construct an arc-analytic function (i.e. a function analytic on each analytic arc) whose graph is not subanalytic.
2
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An arc-analytic function with nondiscrete singular set

100%
Kurdyka K.
Annales Polonici Mathematici
|
1994
|
tom 59
|
nr 3
251-254
EN
We construct an arc-analytic function (i.e. analytic on every real-analytic arc) in ℝ² which is analytic outside a nondiscrete subset of ℝ².
3
Content available remote

Subanalytic version of Whitney's extension theorem

64%
Kurdyka K., Pawłucki W.
Studia Mathematica
|
1997
|
tom 124
|
nr 3
269-280
EN
For any subanalytic $C^k$-Whitney field (k finite), we construct its subanalytic $C^k$-extension to $ℝ^n$. Our method also applies to other o-minimal structures; e.g., to semialgebraic Whitney fields.
4
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O-minimal version of Whitney's extension theorem

64%
Kurdyka K., Pawłucki W.
Studia Mathematica
|
2014
|
tom 224
|
nr 1
81-96
EN
This is a generalized and improved version of our earlier article [Studia Math. 124 (1997)] on the Whitney extension theorem for subanalytic $𝓒^{p}$-Whitney fields (with p finite). In this new version we consider Whitney fields definable in an arbitrary o-minimal structure on any real closed field R and obtain an extension which is a $𝓒^{p}$-function definable in the same o-minimal structure. The Whitney fields that we consider are defined on any locally closed definable subset of Rⁿ. In such a way, a local version of the theorem is included.
5
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Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

64%
D'Acunto D., Kurdyka K.
Annales Polonici Mathematici
|
2005
|
tom 87
|
nr 1
51-61
EN
Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz's gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that $|∇f| ≥ C|f|^{ϱ}$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1 - R(n,d)^{-1}$ with $R(n,d) = d(3d-3)^{n-1}$.
6
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Sum of squares and the Łojasiewicz exponent at infinity

51%
Kurdyka K., Osińska-Ulrych B., Skalski G., Spodzieja S.
Annales Polonici Mathematici
|
2014
|
tom 112
|
nr 3
223-237
EN
Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations $h₁(x) = ⋯ = h_{r}(x) = 0$ and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then $f|_{V}$ extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial $h(x) = ∑_{i=1}^{r} h²_{i}(x)σ_{i}(x)$, where $σ_{i}$ are sums of squares of polynomials of degree at most p, such that f(x) + h(x) > 0 for x ∈ ℝⁿ. We give an estimate for p in terms of: the degree of f, the degrees of $h_{i}$ and the Łojasiewicz exponent at infinity of $f|_{V}$. We prove a version of the above result for polynomials positive on semialgebraic sets. We also obtain a nonnegative extension of some odd power of f which is nonnegative on an irreducible algebraic set.
7
Content available remote

An inverse mapping theorem for arc-analytic homeomorphisms

51%
Fukui T., Kurdyka K., Paunescu L.
Banach Center Publications
|
2004
|
tom 65
|
nr 1
49-56
EN
We show that a subanalytic map-germ (Rⁿ,0) → (Rⁿ,0) which is arc-analytic and bi-Lipschitz has an arc-analytic inverse.
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