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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.

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