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Bi-Lipschitz trivialization of the distance function to a stratum of a stratification

100%

Parusiński A.

Annales Polonici Mathematici

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2005

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tom 87

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nr 1

213-218

EN

Given a Lipschitz stratification 𝒳 that additionally satisfies condition (δ) of Bekka-Trotman (for instance any Lipschitz stratification of a subanalytic set), we show that for every stratum N of 𝒳 the distance function to N is locally bi-Lipschitz trivial along N. The trivialization is obtained by integration of a Lipschitz vector field.

2

A note on singularities at infinity of complex polynomials

100%

Parusiński A.

Banach Center Publications

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1997

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tom 39

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nr 1

131-141

EN

Let f be a complex polynomial. We relate the behaviour of f "at infinity" to the sheaf of vanishing cycles of the family $\overline f$ of projective closures of fibres of f. We show that the absence of such cycles: (i) is equivalent to a condition on the asymptotic behaviour of gradient of f known as Malgrange's Condition, (ii) implies the $C^\infty$-triviality of f. If the support of sheaf of vanishing cycles of $\overline f$ is a finite set, then it detects precisely the change of the topology of the fibres of f. Moreover, in this case, the generic fibre of f has the homotopy type of a bouquet of spheres.

3

On the Euler characteristic of fibres of real polynomial maps

64%

Parusiński A., Szafraniec Z.

Banach Center Publications

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1998

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tom 44

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nr 1

175-182

EN

Let Y be a real algebraic subset of $𝑹^m$ and $F:Y → 𝑹^n$ be a polynomial map. We show that there exist real polynomial functions $g_1, ..., g_s$ on $𝑹^n$ such that the Euler characteristic of fibres of $F$ is the sum of signs of $g_i$.

4

Invariants of bi-Lipschitz equivalence of real analytic functions

64%

Henry J.-P., Parusiński A.

Banach Center Publications

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2004

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tom 65

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nr 1

67-75

EN

We construct an invariant of the bi-Lipschitz equivalence of analytic function germs (ℝⁿ,0) → (ℝ,0) that varies continuously in many analytic families. This shows that the bi-Lipschitz equivalence of analytic function germs admits continuous moduli. For a germ f the invariant is given in terms of the leading coefficients of the asymptotic expansions of f along the sets where the size of |x| |grad f(x)| is comparable to the size of |f(x)|.

5

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Lipschitz stratification of real analytic sets

63%

Parusiński A.

Banach Center Publications

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1988

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tom 20

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nr 1

323-333

Ograniczanie wyników

4 Banach Center Publications

1 Annales Polonici Mathematici

5 Parusiński A.

1 Henry J.-P.

1 Szafraniec Z.

1 2005

1 2004

1 1998

1 1997

1 1988

Bi-Lipschitz trivialization of the distance function to a stratum of a stratification

A note on singularities at infinity of complex polynomials

On the Euler characteristic of fibres of real polynomial maps

Invariants of bi-Lipschitz equivalence of real analytic functions

Lipschitz stratification of real analytic sets