Topological classification of irreducible plane curve singularities in terms of Weierstrass polynomials

Kang, Chunghyuk

doi:10.1090/S0002-9939-1995-1291777-2

Topological classification of irreducible plane curve singularities in terms of Weierstrass polynomials
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by Chunghyuk Kang

Proc. Amer. Math. Soc. 123 (1995), 1363-1371

DOI: https://doi.org/10.1090/S0002-9939-1995-1291777-2

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Abstract:

Let $f(z,y)$ be analytically irreducible at 0 and $f(0) = 0$. Then the plane curve singularity defined by f has the same topological type as the curve defined by ${f_{k + 1}}$ for some $k \geq 0$ where ${f_1} = {z^a} + {y^b},{f_2} = {f_1}^{{n_{21}}} + {y^{{m_{11}}}}{z^{{m_{12}}}},{f_3} = {f_2}^{{n_{31}}} + {f_1}^{{n_{22}}}{y^{{m_{21}}}}{z^{{m_{22}}}}, \ldots$ are defined by induction on k with distinct numerical conditions topologically invariant. Moreover, we give an easy alternate proof of Zariski’s topological classification theorem of irreducible plane curve singularities.

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