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Intersection of analytic curves

Tadeusz KrasińskiKrzysztof Jan Nowak — 2003

Annales Polonici Mathematici

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.

The Łojasiewicz numbers and plane curve singularities

Evelia García BarrosoTadeusz KrasińskiArkadiusz Płoski — 2005

Annales Polonici Mathematici

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

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