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Analytic equivalence of plane curve singularities \(y^n+x^\alpha y+x^\beta a(x)=0\). (English) Zbl 1222.14006
Summary: There are not many examples of complete analytical classifications of specific families of singularities, even in the case of plane algebraic curves. In 1989, C. Kang and S. M. Kim [J. Korean Math. Soc. 26, No. 2, 181–188 (1989; Zbl 0701.32009)] published a paper on analytical classification of plane curve singularities \(y^{n}+a(x)y+b(x)=0\), or, equivalently, \(y^{n}+x^{\alpha}y+x^{\beta}A(x)=0\) where \(A(x)\) is a unit in \(\mathbb{C}\{x\}\), \(\alpha\) and \(\beta\) are integers, \(\alpha\geq n-1\) and \(\beta\geq n\). The classification was not complete in the most difficult case \(\frac{\alpha}{n-1}=\frac{\beta}{n}\). In the present paper, the classification is extended also in this case, the proofs are improved and some gaps are removed.

MSC:
14B05 Singularities in algebraic geometry
14H20 Singularities of curves, local rings
32S15 Equisingularity (topological and analytic)
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