Jump of the Milnor Number in Linear Deformations with Fixed Order

Abstract

Let $f_{0}$ be a plane curve singularity. We study the jump of the Milnor number of a homogeneous singularity $f_{0}$ of order $n$ in the class of linear deformations $f_{s}=f_{0}+sg$ of $f_0$ with fixed order $ord f_{s}=j<n$ for $s\neq0$, where $g$ is holomorphic and $ord g=j$.