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4 Annales Polonici Mathematici

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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}.

Intersection of analytic curves

100%

Krasiński T. , Nowak K.

2003

tom 80

nr 1

193-202

We give a relation between two theories of improper intersections, of Tworzewski and of Stückrad-Vogel, for the case of algebraic curves. Given two arbitrary quasiprojective curves V₁,V₂, the intersection cycle V₁ ∙ V₂ in the sense of Tworzewski turns out to be the rational part of the Vogel cycle v(V₁,V₂). We also give short proofs of two known effective formulae for the intersection cycle V₁ ∙ V₂ in terms of local parametrizations of the curves.

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 ${z ∈ ℂ^n: f₁(z) ·...· fₘ(z) = 0}$.

The Łojasiewicz numbers and plane curve singularities

80%

Barroso E. , Krasiński T. , Płoski A.

nr 1

127-150

For every holomorphic function in two complex variables with an isolated critical point at the origin we consider the Łojasiewicz exponent 𝓛₀(f) defined to be the smallest θ > 0 such that $|grad f(z)| ≥ c|z|^{θ}$ near 0 ∈ ℂ² for some c > 0. We investigate the set of all numbers 𝓛₀(f) where f runs over all holomorphic functions with an isolated critical point at 0 ∈ ℂ².