Arkadiusz Płoski

THE LOJASIEWICZ EXPONENT OF AN ISOLATED WEIGHTED HOMOGENEOUS SURFACE SINGULARITY

Mei-Chi Shaw) We give an explicit formula for the Lojasiewicz exponent of an isolated weighted homogeneous surface singularity in terms of its weights. From the formula we get that the Lojasiewicz exponent is a topological invariant of these singularities.

Polar invariants of plane curve singularities: Intersection theoretical approach

This article, based on the talk given by one of the authors at the Pierrettefest in Castro Urdiales in June 2008, is an overview of a number of recent results on the polar invariants of plane curve singularities.

On the Łojasiewicz numbers

Let f be a holomorphic function of two complex variables with an isolated critical point at 0∈C2. We give some necessary conditions for a rational number to be the smallest θ>0 in the Łojasiewicz inequality |gradf(z)|⩾C|z|θ for z near 0∈C2. To cite this article: E. Garcia Barroso, A. Ploski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

On the Abhyankar-Moh inequality

Abhyankar and Moh in their fundamental paper on the embeddings of the line in the plane proved an important inequality which can be stated in terms of the semigroup associated with the branch at infinity of a plane algebraic curve. In this note we study the semigroups of integers satisfying the Abhyankar-Moh inequality and give a simple proof of the Abhyankar-Moh embedding theorem.

On the Milnor Formula in Arbitrary Characteristic

The Milnor formula μ = 2δ − r + 1 relates the Milnor number μ, the double point number δ and the number r of branches of a plane curve singularity. It holds over the fields of characteristic zero. Melle and Wall based on a result by Deligne proved the inequality μ ≥ 2δ − r + 1 in arbitrary characteristic and showed that the equality μ = 2δ − r + 1 characterizes the singularities with no wild vanishing cycles. In this note we give an account of results on the Milnor formula in characteristic p. It holds if the plane singularity is Newton non-degenerate (Boubakri et al. Rev. Mat. Complut. 25:61–85, 2010) or if p is greater than the intersection number of the singularity with its generic polar (Nguyen Annales de l’Institut Fourier, Tome 66(5):2047–2066, 2016). Then we improve our result on the Milnor number of irreducible singularities (Bull. Lond. Math. Soc. 48:94–98, 2016). Our considerations are based on the properties of polars of plane singularities in characteristic p.

Euclidean algorithm and polynomial equations after Labatie

We recall Labaties effective method of solving polynomial equations with two unknowns by using the Euclidean algorithm.

Invariants of plane curve singularities and Newton diagrams

We present an intersection-theoretical approach to the invariants of plane curve singularities

On the approximate roots of polynomials

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An approach to plane algebroid branches

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Characterization of non-degenerate plane curve singularities

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