Analytic Equivalence of Plane Curve Singularities $y^{n}+x^{\alpha}y+x^{\beta}a(x)=0$

A. Lipkovski, V. Stepanović

There are not many examples of complete analytical classification of specific families of singularities, even in the case of plane algebraic curves. In 1989, Kang and Kim 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 $\mathbb{C}t\{x\}$, $\alpha$ 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.