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A counterexample to subanalyticity of an arc-analytic function

100%

Kurdyka K.

Annales Polonici Mathematici

1991

tom 55

nr 1

241-243

We construct an arc-analytic function (i.e. a function analytic on each analytic arc) whose graph is not subanalytic.

An arc-analytic function with nondiscrete singular set

100%

Kurdyka K.

Annales Polonici Mathematici

1994

tom 59

nr 3

251-254

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.

O-minimal version of Whitney's extension theorem

63%

Kurdyka K. , Pawłucki W.

Studia Mathematica

2014

tom 224

nr 1

81-96

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.

Subanalytic version of Whitney's extension theorem

63%

Kurdyka K. , Pawłucki W.

Studia Mathematica

1997

tom 124

nr 3

269-280

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.

Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

63%

D'Acunto D. , Kurdyka K.

Annales Polonici Mathematici

2005

tom 87

nr 1

51-61

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 $R(n,d) = d(3d-3)^{n-1}$.

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

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.