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A set on which the local Łojasiewicz exponent is attained

100%

Chądzyński J. , Krasiński T.

1997

tom 67

nr 3

297-301

Let U be a neighbourhood of 0 ∈ ℂⁿ. We show that for a holomorphic mapping $F = (f₁,..., fₘ): U → ℂ^m$, F(0) = 0, the Łojasiewicz exponent 𝓛₀(F) is attained on the set {z ∈ U: f₁(z)·...·fₘ(z) = 0}.

A set on which the Łojasiewicz exponent at infinity is attained

100%

Chądzyński J. , Krasiński T.

1997

tom 67

nr 2

191-197

We show that for a polynomial mapping $F = (f₁,..., fₘ): ℂ^n → ℂ^m$ the Łojasiewicz exponent $𝓛_∞(F)$ of F is attained on the set ${z ∈ ℂ^n: f₁(z) ·...· fₘ(z) = 0}$.

On the Łojasiewicz exponent near the fibre of a polynomial

80%

Skalski G.

tom Vol. 52, nr 3

231-236

The equivalence of the definitions of the Łojasiewicz exponent introduced by Ha and by Chądzyński and Krasiński is proved. Moreover we show that if the above exponents are less than -1 then they are attained at a curve meromorphic at infinity.

The Fedoryuk condition and the Łojasiewicz exponent near a fibre of a polynomial

80%

Rodak T.

tom Vol. 52, nr 3

227-229

We give a description of the set of points for which the Fedoryuk condition fails in terms of the Łojasiewicz exponent at infinity near a fibre of a polynomial.

Multiplicity and Semicontinuity of the Łojasiewicz Exponent

80%

Rodak T. , Różycki A. , Spodzieja S.

tom Vol. 64, no. 1

55--62

We give an effective formula for the improper isolated multiplicity of a polynomial mapping. Using this formula we construct, for a given deformation of a holomorphic mapping with an isolated zero at zero, a stratification of the space of parameters such that the Łojasiewicz exponent is constant on each stratum.